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Unit 1: Get ready for equations & inequalities
About this unit.
Tackle equations and inequalities with confidence! In this unit, we'll learn how to check your work, spot errors, and use key properties to simplify and solve problems.
Algebraic equations basics
- Variables, expressions, & equations (Opens a modal)
- Testing solutions to equations (Opens a modal)
- Intro to equations (Opens a modal)
- Testing solutions to equations Get 5 of 7 questions to level up!
One-step addition & subtraction equations
- One-step addition & subtraction equations (Opens a modal)
- One-step addition equation (Opens a modal)
- One-step addition & subtraction equations Get 5 of 7 questions to level up!
One-step multiplication & division equations
- One-step division equations (Opens a modal)
- One-step multiplication equations (Opens a modal)
- One-step multiplication & division equations (Opens a modal)
- One-step multiplication & division equations Get 5 of 7 questions to level up!
Finding mistakes in one-step equations
- Finding mistakes in one-step equations (Opens a modal)
- Find the mistake in one-step equations Get 3 of 4 questions to level up!
Intro to inequalities with variables
- Plotting inequalities on a number line (Opens a modal)
- Plotting an inequality example (Opens a modal)
- Inequality from graph Get 3 of 4 questions to level up!
- Plotting inequalities Get 3 of 4 questions to level up!
Testing solutions to inequalities
- Testing solutions to inequalities (Opens a modal)
- Testing solutions to inequalities Get 3 of 4 questions to level up!
Combining like terms
- Intro to combining like terms (Opens a modal)
- Combining like terms with negative coefficients (Opens a modal)
- Combining like terms with rational coefficients (Opens a modal)
- Combining like terms with negative coefficients Get 5 of 7 questions to level up!
- Combining like terms with rational coefficients Get 3 of 4 questions to level up!
The distributive property & equivalent expressions
- The distributive property with variables (Opens a modal)
- Factoring with the distributive property (Opens a modal)
- Combining like terms with negative coefficients & distribution (Opens a modal)
- Equivalent expressions: negative numbers & distribution (Opens a modal)
- Distributive property with variables (negative numbers) Get 3 of 4 questions to level up!
- Combining like terms with negative coefficients & distribution Get 3 of 4 questions to level up!
- Equivalent expressions: negative numbers & distribution Get 5 of 7 questions to level up!
Two-step equations
- Same thing to both sides of equations (Opens a modal)
- Intro to two-step equations (Opens a modal)
- Two-step equations intuition (Opens a modal)
- Worked example: two-step equations (Opens a modal)
- Two-step equations Get 5 of 7 questions to level up!
Finding mistakes in two-step equations
- Find the mistake: two-step equations (Opens a modal)
- Two-step equations review (Opens a modal)
- Find the mistake: two-step equations Get 3 of 4 questions to level up!
One- and two-step inequalities
- One-step inequalities examples (Opens a modal)
- One-step inequalities: -5c ≤ 15 (Opens a modal)
- Two-step inequalities (Opens a modal)
- One-step inequalities Get 5 of 7 questions to level up!
- Two-step inequalities Get 5 of 7 questions to level up!
Equations with variables on both sides
- Intro to equations with variables on both sides (Opens a modal)
- Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
- Equations with variables on both sides Get 3 of 4 questions to level up!
Equations with parentheses
- Equations with parentheses (Opens a modal)
- Multi-step equations review (Opens a modal)
- Equations with parentheses Get 3 of 4 questions to level up!
2.1 The Rectangular Coordinate Systems and Graphs
x -intercept is ( 4 , 0 ) ; ( 4 , 0 ) ; y- intercept is ( 0 , 3 ) . ( 0 , 3 ) .
125 = 5 5 125 = 5 5
( − 5 , 5 2 ) ( − 5 , 5 2 )
2.2 Linear Equations in One Variable
x = −5 x = −5
x = −3 x = −3
x = 10 3 x = 10 3
x = 1 x = 1
x = − 7 17 . x = − 7 17 . Excluded values are x = − 1 2 x = − 1 2 and x = − 1 3 . x = − 1 3 .
x = 1 3 x = 1 3
m = − 2 3 m = − 2 3
y = 4 x −3 y = 4 x −3
x + 3 y = 2 x + 3 y = 2
Horizontal line: y = 2 y = 2
Parallel lines: equations are written in slope-intercept form.
y = 5 x + 3 y = 5 x + 3
2.3 Models and Applications
C = 2.5 x + 3 , 650 C = 2.5 x + 3 , 650
L = 37 L = 37 cm, W = 18 W = 18 cm
2.4 Complex Numbers
−24 = 0 + 2 i 6 −24 = 0 + 2 i 6
( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i ( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i
5 2 − i 5 2 − i
18 + i 18 + i
−3 −4 i −3 −4 i
2.5 Quadratic Equations
( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1 ( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1
( x −7 ) ( x + 3 ) = 0 , ( x −7 ) ( x + 3 ) = 0 , x = 7 , x = 7 , x = −3. x = −3.
( x + 5 ) ( x −5 ) = 0 , ( x + 5 ) ( x −5 ) = 0 , x = −5 , x = −5 , x = 5. x = 5.
( 3 x + 2 ) ( 4 x + 1 ) = 0 , ( 3 x + 2 ) ( 4 x + 1 ) = 0 , x = − 2 3 , x = − 2 3 , x = − 1 4 x = − 1 4
x = 0 , x = −10 , x = −1 x = 0 , x = −10 , x = −1
x = 4 ± 5 x = 4 ± 5
x = 3 ± 22 x = 3 ± 22
x = − 2 3 , x = − 2 3 , x = 1 3 x = 1 3
2.6 Other Types of Equations
{ −1 } { −1 }
0 , 0 , 1 2 , 1 2 , − 1 2 − 1 2
1 ; 1 ; extraneous solution − 2 9 − 2 9
−2 ; −2 ; extraneous solution −1 −1
−1 , −1 , 3 2 3 2
−3 , 3 , − i , i −3 , 3 , − i , i
2 , 12 2 , 12
−1 , −1 , 0 0 is not a solution.
2.7 Linear Inequalities and Absolute Value Inequalities
[ −3 , 5 ] [ −3 , 5 ]
( − ∞ , −2 ) ∪ [ 3 , ∞ ) ( − ∞ , −2 ) ∪ [ 3 , ∞ )
x < 1 x < 1
x ≥ −5 x ≥ −5
( 2 , ∞ ) ( 2 , ∞ )
[ − 3 14 , ∞ ) [ − 3 14 , ∞ )
6 < x ≤ 9 or ( 6 , 9 ] 6 < x ≤ 9 or ( 6 , 9 ]
( − 1 8 , 1 2 ) ( − 1 8 , 1 2 )
| x −2 | ≤ 3 | x −2 | ≤ 3
k ≤ 1 k ≤ 1 or k ≥ 7 ; k ≥ 7 ; in interval notation, this would be ( − ∞ , 1 ] ∪ [ 7 , ∞ ) . ( − ∞ , 1 ] ∪ [ 7 , ∞ ) .
2.1 Section Exercises
Answers may vary. Yes. It is possible for a point to be on the x -axis or on the y -axis and therefore is considered to NOT be in one of the quadrants.
The y -intercept is the point where the graph crosses the y -axis.
The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , 6 ) . ( 0 , 6 ) .
The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , −3 ) . ( 0 , −3 ) .
The x- intercept is ( 3 , 0 ) ( 3 , 0 ) and the y -intercept is ( 0 , 9 8 ) . ( 0 , 9 8 ) .
y = 4 − 2 x y = 4 − 2 x
y = 5 − 2 x 3 y = 5 − 2 x 3
y = 2 x − 4 5 y = 2 x − 4 5
d = 74 d = 74
d = 36 = 6 d = 36 = 6
d ≈ 62.97 d ≈ 62.97
( 3 , − 3 2 ) ( 3 , − 3 2 )
( 2 , −1 ) ( 2 , −1 )
( 0 , 0 ) ( 0 , 0 )
y = 0 y = 0
not collinear
A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 ) A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 )
d = 8.246 d = 8.246
d = 5 d = 5
( −3 , 4 ) ( −3 , 4 )
x = 0 y = −2 x = 0 y = −2
x = 0.75 y = 0 x = 0.75 y = 0
x = − 1.667 y = 0 x = − 1.667 y = 0
15 − 11.2 = 3.8 mi 15 − 11.2 = 3.8 mi shorter
6 .0 42 6 .0 42
Midpoint of each diagonal is the same point ( 2 , –2 ) ( 2 , –2 ) . Note this is a characteristic of rectangles, but not other quadrilaterals.
2.2 Section Exercises
It means they have the same slope.
The exponent of the x x variable is 1. It is called a first-degree equation.
If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).
x = 2 x = 2
x = 2 7 x = 2 7
x = 6 x = 6
x = 3 x = 3
x = −14 x = −14
x ≠ −4 ; x ≠ −4 ; x = −3 x = −3
x ≠ 1 ; x ≠ 1 ; when we solve this we get x = 1 , x = 1 , which is excluded, therefore NO solution
x ≠ 0 ; x ≠ 0 ; x = − 5 2 x = − 5 2
y = − 4 5 x + 14 5 y = − 4 5 x + 14 5
y = − 3 4 x + 2 y = − 3 4 x + 2
y = 1 2 x + 5 2 y = 1 2 x + 5 2
y = −3 x − 5 y = −3 x − 5
y = 7 y = 7
y = −4 y = −4
8 x + 5 y = 7 8 x + 5 y = 7
Perpendicular
m = − 9 7 m = − 9 7
m = 3 2 m = 3 2
m 1 = − 1 3 , m 2 = 3 ; Perpendicular . m 1 = − 1 3 , m 2 = 3 ; Perpendicular .
y = 0.245 x − 45.662. y = 0.245 x − 45.662. Answers may vary. y min = −50 , y max = −40 y min = −50 , y max = −40
y = − 2.333 x + 6.667. y = − 2.333 x + 6.667. Answers may vary. y min = −10 , y max = 10 y min = −10 , y max = 10
y = − A B x + C B y = − A B x + C B
The slope for ( −1 , 1 ) to ( 0 , 4 ) is 3. The slope for ( −1 , 1 ) to ( 2 , 0 ) is − 1 3 . The slope for ( 2 , 0 ) to ( 3 , 3 ) is 3. The slope for ( 0 , 4 ) to ( 3 , 3 ) is − 1 3 . The slope for ( −1 , 1 ) to ( 0 , 4 ) is 3. The slope for ( −1 , 1 ) to ( 2 , 0 ) is − 1 3 . The slope for ( 2 , 0 ) to ( 3 , 3 ) is 3. The slope for ( 0 , 4 ) to ( 3 , 3 ) is − 1 3 .
Yes they are perpendicular.
2.3 Section Exercises
Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading.
2 , 000 − x 2 , 000 − x
v + 10 v + 10
Ann: 23 ; 23 ; Beth: 46 46
20 + 0.05 m 20 + 0.05 m
90 + 40 P 90 + 40 P
50 , 000 − x 50 , 000 − x
She traveled for 2 h at 20 mi/h, or 40 miles.
$5,000 at 8% and $15,000 at 12%
B = 100 + .05 x B = 100 + .05 x
R = 9 R = 9
r = 4 5 r = 4 5 or 0.8
W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14 W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14
f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21 f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21
m = − 5 4 m = − 5 4
h = 2 A b 1 + b 2 h = 2 A b 1 + b 2
length = 360 ft; width = 160 ft
A = 88 in . 2 A = 88 in . 2
h = V π r 2 h = V π r 2
r = V π h r = V π h
C = 12 π C = 12 π
2.4 Section Exercises
Add the real parts together and the imaginary parts together.
Possible answer: i i times i i equals -1, which is not imaginary.
−8 + 2 i −8 + 2 i
14 + 7 i 14 + 7 i
− 23 29 + 15 29 i − 23 29 + 15 29 i
8 − i 8 − i
−11 + 4 i −11 + 4 i
2 −5 i 2 −5 i
6 + 15 i 6 + 15 i
−16 + 32 i −16 + 32 i
−4 −7 i −4 −7 i
2 − 2 3 i 2 − 2 3 i
4 − 6 i 4 − 6 i
2 5 + 11 5 i 2 5 + 11 5 i
1 + i 3 1 + i 3
( 3 2 + 1 2 i ) 6 = −1 ( 3 2 + 1 2 i ) 6 = −1
5 −5 i 5 −5 i
9 2 − 9 2 i 9 2 − 9 2 i
2.5 Section Exercises
It is a second-degree equation (the highest variable exponent is 2).
We want to take advantage of the zero property of multiplication in the fact that if a ⋅ b = 0 a ⋅ b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0 o r b = 0. a = 0 o r b = 0.
One, when no linear term is present (no x term), such as x 2 = 16. x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d . ( a x + b ) 2 = d .
x = 6 , x = 6 , x = 3 x = 3
x = − 5 2 , x = − 5 2 , x = − 1 3 x = − 1 3
x = 5 , x = 5 , x = −5 x = −5
x = − 3 2 , x = − 3 2 , x = 3 2 x = 3 2
x = −2 , 3 x = −2 , 3
x = 0 , x = 0 , x = − 3 7 x = − 3 7
x = −6 , x = −6 , x = 6 x = 6
x = 6 , x = 6 , x = −4 x = −4
x = 1 , x = 1 , x = −2 x = −2
x = −2 , x = −2 , x = 11 x = 11
z = 2 3 , z = 2 3 , z = − 1 2 z = − 1 2
x = 3 ± 17 4 x = 3 ± 17 4
One rational
Two real; rational
x = − 1 ± 17 2 x = − 1 ± 17 2
x = 5 ± 13 6 x = 5 ± 13 6
x = − 1 ± 17 8 x = − 1 ± 17 8
x ≈ 0.131 x ≈ 0.131 and x ≈ 2.535 x ≈ 2.535
x ≈ − 6.7 x ≈ − 6.7 and x ≈ 1.7 x ≈ 1.7
a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a
x ( x + 10 ) = 119 ; x ( x + 10 ) = 119 ; 7 ft. and 17 ft.
maximum at x = 70 x = 70
The quadratic equation would be ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. The two values of x x are 20 and 60.
2.6 Section Exercises
This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.
He or she is probably trying to enter negative 9, but taking the square root of −9 −9 is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in −27. −27.
A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.
x = 81 x = 81
x = 17 x = 17
x = 8 , x = 27 x = 8 , x = 27
x = −2 , 1 , −1 x = −2 , 1 , −1
y = 0 , 3 2 , − 3 2 y = 0 , 3 2 , − 3 2
m = 1 , −1 m = 1 , −1
x = 2 5 , ±3 i x = 2 5 , ±3 i
x = 32 x = 32
t = 44 3 t = 44 3
x = −2 x = −2
x = 4 , −4 3 x = 4 , −4 3
x = − 5 4 , 7 4 x = − 5 4 , 7 4
x = 3 , −2 x = 3 , −2
x = 1 , −1 , 3 , -3 x = 1 , −1 , 3 , -3
x = 2 , −2 x = 2 , −2
x = 1 , 5 x = 1 , 5
x ≥ 0 x ≥ 0
x = 4 , 6 , −6 , −8 x = 4 , 6 , −6 , −8
2.7 Section Exercises
When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.
( − ∞ , ∞ ) ( − ∞ , ∞ )
We start by finding the x -intercept, or where the function = 0. Once we have that point, which is ( 3 , 0 ) , ( 3 , 0 ) , we graph to the right the straight line graph y = x −3 , y = x −3 , and then when we draw it to the left we plot positive y values, taking the absolute value of them.
( − ∞ , 3 4 ] ( − ∞ , 3 4 ]
[ − 13 2 , ∞ ) [ − 13 2 , ∞ )
( − ∞ , 3 ) ( − ∞ , 3 )
( − ∞ , − 37 3 ] ( − ∞ , − 37 3 ]
All real numbers ( − ∞ , ∞ ) ( − ∞ , ∞ )
( − ∞ , − 10 3 ) ∪ ( 4 , ∞ ) ( − ∞ , − 10 3 ) ∪ ( 4 , ∞ )
( − ∞ , −4 ] ∪ [ 8 , + ∞ ) ( − ∞ , −4 ] ∪ [ 8 , + ∞ )
No solution
( −5 , 11 ) ( −5 , 11 )
[ 6 , 12 ] [ 6 , 12 ]
[ −10 , 12 ] [ −10 , 12 ]
x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 , ( − 2 , + ∞ ) x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 , ( − 2 , + ∞ )
x < − 3 or x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ [ 1 , ∞ ) x < − 3 or x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ [ 1 , ∞ )
( − ∞ , −1 ) ∪ ( 3 , ∞ ) ( − ∞ , −1 ) ∪ ( 3 , ∞ )
[ −11 , −3 ] [ −11 , −3 ]
It is never less than zero. No solution.
Where the blue line is above the orange line; point of intersection is x = − 3. x = − 3.
( − ∞ , −3 ) ( − ∞ , −3 )
Where the blue line is above the orange line; always. All real numbers.
( − ∞ , − ∞ ) ( − ∞ , − ∞ )
( −1 , 3 ) ( −1 , 3 )
( − ∞ , 4 ) ( − ∞ , 4 )
{ x | x < 6 } { x | x < 6 }
{ x | −3 ≤ x < 5 } { x | −3 ≤ x < 5 }
( −2 , 1 ] ( −2 , 1 ]
( − ∞ , 4 ] ( − ∞ , 4 ]
Where the blue is below the orange; always. All real numbers. ( − ∞ , + ∞ ) . ( − ∞ , + ∞ ) .
Where the blue is below the orange; ( 1 , 7 ) . ( 1 , 7 ) .
x = 2 , − 4 5 x = 2 , − 4 5
( −7 , 5 ] ( −7 , 5 ]
80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400 80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400
[ 1 , 600 , 2 , 400 ] [ 1 , 600 , 2 , 400 ]
Review Exercises
x -intercept: ( 3 , 0 ) ; ( 3 , 0 ) ; y -intercept: ( 0 , −4 ) ( 0 , −4 )
y = 5 3 x + 4 y = 5 3 x + 4
72 = 6 2 72 = 6 2
620.097 620.097
midpoint is ( 2 , 23 2 ) ( 2 , 23 2 )
x = 4 x = 4
x = 12 7 x = 12 7
y = 1 6 x + 4 3 y = 1 6 x + 4 3
y = 2 3 x + 6 y = 2 3 x + 6
females 17, males 56
x = − 3 4 ± i 47 4 x = − 3 4 ± i 47 4
horizontal component −2 ; −2 ; vertical component −1 −1
7 + 11 i 7 + 11 i
−16 − 30 i −16 − 30 i
−4 − i 10 −4 − i 10
x = 7 − 3 i x = 7 − 3 i
x = −1 , −5 x = −1 , −5
x = 0 , 9 7 x = 0 , 9 7
x = 10 , −2 x = 10 , −2
x = − 1 ± 5 4 x = − 1 ± 5 4
x = 2 5 , − 1 3 x = 2 5 , − 1 3
x = 5 ± 2 7 x = 5 ± 2 7
x = 0 , 256 x = 0 , 256
x = 0 , ± 2 x = 0 , ± 2
x = 11 2 , −17 2 x = 11 2 , −17 2
[ − 10 3 , 2 ] [ − 10 3 , 2 ]
( − 4 3 , 1 5 ) ( − 4 3 , 1 5 )
Where the blue is below the orange line; point of intersection is x = 3.5. x = 3.5.
( 3.5 , ∞ ) ( 3.5 , ∞ )
Practice Test
y = 3 2 x + 2 y = 3 2 x + 2
( 0 , −3 ) ( 0 , −3 ) ( 4 , 0 ) ( 4 , 0 )
( − ∞ , 9 ] ( − ∞ , 9 ]
x = −15 x = −15
x ≠ −4 , 2 ; x ≠ −4 , 2 ; x = − 5 2 , 1 x = − 5 2 , 1
x = 3 ± 3 2 x = 3 ± 3 2
( −4 , 1 ) ( −4 , 1 )
y = −5 9 x − 2 9 y = −5 9 x − 2 9
y = 5 2 x − 4 y = 5 2 x − 4
5 13 − 14 13 i 5 13 − 14 13 i
x = 2 , − 4 3 x = 2 , − 4 3
x = 1 2 ± 2 2 x = 1 2 ± 2 2
x = 1 2 , 2 , −2 x = 1 2 , 2 , −2
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Math With Mrs. Molina
There are two things we must give children: the first one is roots and the other wings., unit 3: equations and inequalities, click here to go to the ixl website for all kinds of 8th grade topics and review problems..
What is an equation?
Examples: 4 + 3 = 7 or 3x + 5 = 10
An equation is a number sentence. We call it an equation because it has an equal sign.
The 5 Steps to Writing an Equation or Inequality
Step 1. read and underline the question, step 2. find your χ (your variable/unknown) and box it, step 3. circle the math words (product, quotient, each, per, together, sum, difference, squared ), step 4 . replace the operation words with their symbols ( • , + , – , ÷ , / , = , < , > , ≤ , ≥ ,√ , ≠ , ² , ³ ), step 5. write the equations.
Don’t forget our cool ‘dance’ we did to remember this!
WRITING EQUATIONS PRACTICE PROBLEMS:
Click here to practice Writing Equations online and get automatic feedback (it grades it)! 🙂
With Equations, Inequalities and Expressions we always want to combine like terms 1st!
Here is an example on how to do that:
Once all like terms have been combined then we can solve.
Solving Equations with Models
To create your own equations using models click here !
MODELING EQUATIONS PRACTICE PROBLEMS:
Click here to practice Modeling Equations online and get automatic feedback (it grades it)! 🙂
Solving Equations Algebraically
Here is another example solving algebraically, solve √(x/2) = 3.
And the more “tricks” and techniques you learn the better you will get.
Here is an example of how we solved equations in class:
SOLVING EQUATIONS (with variables on both sides PRACTICE PROBLEMS:
Click here or here to practice Solving Equations online and get automatic feedback (it grades it)! 🙂
Systems of Equations
For information on systems of equations click here ., simple vs. compound interest, introduction to interest :.
http://www.mathsisfun.com/money/interest.html
SIMPLE INTEREST
I = Prt
- I = interest owed [$] (this is ONLY the interest borrowed)
- P = amount borrowed (called “Principal”) [$]
- r = interest rate [%] (you have to divide the percent by 100) For information On Percents click here !
- t = time [years]
Simple interest is money you can earn by investing some money (the principal). The interest (percent) is the rate that makes the money grow!
COMPOUND INTEREST
A = P(1+r)^t
- A = All of it / Actual / total amount owed (this amount includes the interest and the principal) [$]
- P = amount borrowed (called “Principal”) [$]
- r = interest rate [%]
Compound interest is very similar to simple interest. The difference is that compound interest grows much faster ! The reason it grows faster is because the interest (percent) has an exponent .
********** MAKE SURE TO READ THE QUESTION AND SEE EXACTLY WHAT IT IS ASKING DOES IT JUST WANT THE INTEREST OR THE TOTAL (All of it) ???????? *************************
For information on compound interest click here.
SIMPLE INTEREST PRACTICE PROBLEMS:
Click here or here to practice Simple Interest online and get automatic feedback (it grades it)! 🙂
COMPOUND INTEREST PRACTICE PROBLEMS:
Click here or here to practice Compound Interest online and get automatic feedback (it grades it)! 🙂
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Equations and Inequalities (Math 7 Curriculum - Unit 3) | All Things Algebra®
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Description
This Equations and Inequalities Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the following topics:
• One-Step Equations with Addition and Subtraction
• One-Step Equations with Multiplication and Division
• One-Step Equations with Rational Numbers (Decimals and Fractions)
• Translating One-Step Equations
• One-Step Equation Word Problems
• Two-Step Equations
• Translating Two-Step Equations
• Two-Step Equations Word Problems
• Multi-Step Equations (Variables on One Side)
• Multi-Step Equations (Variables on Both Sides)
• Writing and Graphing Inequalities
• Solving One-Step Inequalities
• Solving Two-Step Inequalities
• Translating One- and Two-Step Inequalities
• One- and Two-Step Inequality Word Problems
ADDITIONAL COMPONENTS INCLUDED:
(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice. Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.
(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes; this is the PDF in Google Slides. I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead!
How does this unit compare to your Pre-Algebra Equations and Inequalities Unit? This unit is similar in that it contains similar topics. However, all material has been rewritten to ensure there is no duplication. The biggest differences between this Math 7 unit and Pre-Algebra Curriculum include no square roots equations, special solutions, multi-step equation word problems, or multi-step inequalities. The material is also presented at a slighter slower pace.
This resource is included in the following bundle(s):
Math 7 Curriculum
More Math 7 Units:
Unit 1 – Number Sense
Unit 2 – Expressions
Unit 4 – Ratios, Proportions, and Percents
Unit 5 – Functions and Graphing
Unit 6 – Geometry
Unit 7 – Measurement: Area and Volume
Unit 8 – Probability and Statistics
LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].
COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.
© All Things Algebra (Gina Wilson), 2012-present
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Gina Wilson All Things Algebra Answer Key | Gina Wilson All things Algebra 2015
In the realm of mathematics education, finding reliable resources that support effective learning can be a challenging task. However, Gina Wilson All Things Algebra is a comprehensive platform that provides educators and students with valuable tools to enhance their mathematical knowledge. One of the key features of this platform is the availability of the answer key, which serves as a vital resource for learners seeking to validate their solutions and progress in their mathematical journey. In this article, we will delve into the benefits of Gina Wilson All Things Algebra and explore how the answer key can be accessed and utilized effectively.
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What is Gina Wilson All Things Algebra?
Gina Wilson All Things Algebra is an educational platform developed by Gina Wilson, an experienced mathematics educator. It offers a wide range of resources, including curriculum materials, lesson plans, activities, and assessments, designed to promote a deeper understanding of algebraic concepts. The platform caters to both teachers and students, providing them with the necessary tools to excel in algebraic reasoning and problem-solving.
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The Birth of All Things Algebra 2015
All Things Algebra 2015 was born out of Gina Wilson's desire to provide teachers with a comprehensive and easy-to-use curriculum that would help them engage their students and promote deep understanding of mathematical concepts. Recognizing the need for high-quality resources, Gina Wilson set out to create a platform that would serve as a one-stop-shop for educators seeking effective teaching materials.
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3. Is there a free trial available for All Things Algebra 2015?
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Gina Wilson is dedicated to continuous improvement and regularly adds new resources to All Things Algebra 2015. Updates are released periodically to enhance the curriculum and address emerging educational trends.
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The steps to solve an equation with decimals or fractions are exactl the same! Locate the variable. Determine the operation tied to the variable. Use inverse operations on both sides of the equal sign to solve. Check your solutionl Directions: Solve each equation. Check all solutions. Date: Class: -3.15: -z.scl.s) -3-1s -3-1M -1.2 —6.2 + 1.25 ...
difference, minus, less than, subtracted from, decreased by, take away. product, times, multiplied by, of, per, twice, triple, double. The form px+q=r. the relationship between two numbers that are not equal. Uses symbols like less than ( < ) and greater than ( > ) Key words from unit 3 Learn with flashcards, games, and more — for free.
6th grade 11 units · 148 skills. Unit 1 Ratios. Unit 2 Arithmetic with rational numbers. Unit 3 Rates and percentages. Unit 4 Exponents and order of operations. Unit 5 Negative numbers. Unit 6 Variables & expressions. Unit 7 Equations & inequalities. Unit 8 Plane figures.
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!
Name: _____ Unit 3: Equations & Inequalities Date: _____ Per: _____ Homework 3: Two-Step Equations ** Directions: Solve each equation. Check all solutions. 1. 34x −=−31 2. 33 =− +27a 3. 17 9 4 w += − 4. −− =17p 13 5. −− =51k 639 6. 28 26 7 r −=− 7. 14 11 5 n −=− + 8. 26−=f 40 9. 2.81145k −= 10. 19 1.5 r
Unit 3: Systems of Linear Equations and Inequalities. Solve by graphing. Click the card to flip 👆. Method used to solve a system of equations in order to find the point (s) that make BOTH equations true. For this method, each line is graphed on the same coordinate plane. It is the least accurate, if the solution has large or rational numbers.
Select a Unit. Unit 1 Sequences; Unit 2 Linear and Exponential Functions; Unit 3 Features of Functions; Unit 4 Equations and Inequalities; Unit 5 Systems of Equations and Inequalities; Unit 6 Quadratic Functions; Unit 7 Structures of Quadratic Expressions; Unit 8 More Functions, More Features; Unit 9 Modeling Data
Study with Quizlet and memorize flashcards containing terms like What is the distributive property?, What are inverse operations?, How are rational equations solved? and more.
Get ready for Algebra 1 6 units · 87 skills. Unit 1 Get ready for equations & inequalities. Unit 2 Get ready for working with units. Unit 3 Get ready for linear relationships. Unit 4 Get ready for functions & sequences. Unit 5 Get ready for exponents, radicals, & irrational numbers. Unit 6 Get ready for quadratics. Course challenge.
Basic Algebra: Unit 3 Equations and Inequalities Equations and Inequalities • An equation involves an equal sign and indicates that two expressions have the same value. x+42 = 67(4−x) is an equation, and means x+42 has the same value as 67(4−x). • Equivalent equations are equations that have exactly the same solution.
See Answer See Answer See Answer done loading Question: Unit 1: Equations & Inequalities Homework 3: Solving Equations page document! ** 2-3.96-23) 2.-3-9(5-2k) Show transcribed image text
SOC 225, Chapter 13. 10 terms. quizlette373090309. Preview. martin list. 10 terms. martavej22. Preview. Study with Quizlet and memorize flashcards containing terms like Addition Property of Equality, Division Property of Equality, Equation and more.
For questions I -8, solve the equation. Unit 3 Test Equations & Inequalities Show all work and check each solution. 2. 4.5 s. W For questions 9-10, translate the equation using a variable, then solve. Check each solution. 9. "The quotient of number and 10. "Seven subfracted from a number is -18." 2017 Unit Test Study Quide (Equations ...
1.2 Section Exercises. 1. No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 23 2 3 is the same as 2 × 2 × 2, 2 × 2 × 2, which is 8. 32 3 2 is the same as 3 × 3, 3 × 3, which is 9. 3. It is a method of writing very small and very large numbers. 5.
Unit 5 - Systems of Equations & Inequalities (Updated October 2016) copy. Name: Date: Unit 5: Systems of Equations & Inequalities Homework 1: Solving Systems by Graphing ** This is a 2-page document! ** Solve each system of equations by graphing. Clearly identify your solution. -16 — 6y = 30 9x + = 12 +4 v = —12 O Gina Wilson (All Things ...
This Equations and Inequalities Unit Bundle includes guided notes, homework assignments, four quizzes, a study guide, and a unit test that cover the following topics: • One-Step Equations. • Rational Equations. • Two-Step Equations. • Solving Equations by Square Roots.
Introduction to Systems of Equations and Inequalities; 7.1 Systems of Linear Equations: Two Variables; 7.2 Systems of Linear Equations: Three Variables; 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 7.4 Partial Fractions; 7.5 Matrices and Matrix Operations; 7.6 Solving Systems with Gaussian Elimination; 7.7 Solving Systems with Inverses; 7.8 Solving Systems with Cramer's Rule
File Size: 323 kb. File Type: pdf. Download File. AP Learning Objectives: 3.10.A Solve equations and inequalities involving trigonometric functions. *AP® is a trademark registered and owned by the CollegeBoard, which was not involved in the production of, and does not endorse, this site.
Equations. What is an equation? Examples: 4 + 3 = 7 or 3x + 5 = 10. An equation is a number sentence. We call it an equation because it has an equal sign. The 5 Steps to Writing an Equation or Inequality Step 1. Read and underline the question Step 2. Find your Χ (your variable/unknown) and BOX it Step 3.
This Equations and Inequalities Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the following topics: • One-Step Equations with Addition and Subtraction. • One-Step Equations with Multiplication and Division. • One-Step Equations with Rational Numbers (Decimals and Fractions)
Worksheets are Unit 3 linear systems, Algebra i, Equations inequalities, Algebra 1 unit 2 equations and inequalities answer key, 8th math unit 3, Unit 1 equations and inequalities answers, Solving inequalities 4 directions x examples, Georgia standards of excellence curriculum frameworks. *Click on Open button to open and print to worksheet.
Only a nontransferable license is available for this resource. This unit includes 62 pages of guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the topics listed in the description below. VIEW PREVIEW. Add to Cart.
The answer key on Gina Wilson All Things Algebra offers various features that enhance the learning experience. Some notable features include: Detailed Solutions: The answer key provides comprehensive and detailed solutions to the exercises, enabling students to identify any errors and learn from them. Multiple Approaches: In many cases, the ...