Download Block: ______ Review for Solving Equations, Inequalities and

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Block: ________
Review for Solving Equations, Inequalities and Proofs Summative
Solving Equations
Solve each equation. Show your work! Leave all answers as integers or
fractions.
2. 5 – 2x = 3x + 8
1. x + (-9) = 13
x - 9 = 13
x = 22
Simplify the equation
Add 9 to both sides
5 = 5x +8
-3 = 5x
x=-
3. ⅔(18x – 12) = 7 – 3(x – 3)
12x – 8 = 7 - 3x + 9
12x – 8 = 16 – 3x
15x – 8 = 16
15x = 24
x=
Distribute
Combine like terms
Add 3x to both sides
Add 8 to both sides
Divide both sides by 15
x=
Simplify
5.
= -5
x = 20
+ 3 = -2
Add 2x to both sides
Subtract 8 from both sides
Divide both sides by 5
4. 5(2 – x) + 7x = -3(x + 5)
10 – 5x + 7x = -3x -15
Distribute
10 + 2x = -3x – 15
Combine like terms
10 + 5x = -15
Add 3x to both sides
5x = -25
Subtract 10 from both sides
x = -5
Divide both sides by 5
6. x = -20
Subtract both sides by 3
x = -25 Multiply both sides by the reciprocal
Multiply both sides by -4
7.
6(d +13) = 18 (d -13)
6d + 78 = 18d – 234
78 = 12d – 234
312 = 12d
d = 26
8.
Cross Multiply
Distribute
Subtract 6d from both sides
Add 234 to both sides
Divide both sides by 12
c – 8 = - 20
c = -12
Multiply by -2
Add 8 to both sides
Solving Inequalities
Solve each inequality. Show your work!
9.
-4
10. x < -39
x< 12 Multiply by -3
x < - 65
Multiply by the reciprocal
Flip the Sign
11. 2(x + 6) < – (2x + 2)
2x + 12 < -2x -2
4x +12 < -2
4x < -14
12. 8(x – 1) > -8 + 8x
8x – 8 > -8 +8x
Distribute
0 – 8 > -8
Subtract 8x from both sides
- 8 > -8
This is a FALSE statement so...
Distribute
Add 2x to both sides
Subtract 12 from both sides
x<
Divide both sides by 4
x<
NO SOLUTION
Simplify the fraction
Solve each compound inequality.
13. -7 < 2x – 1
-6 < 2x < 12
-3 < x < 6
11
4 – 3y
14. -2
Add 1 to both sides
Divide both sides by 2
-6 < -3y < 9
2 > y > -3
Subtract 4 from both sides
Divide both sides by -3 and
flip the sign!
Use Symmetric Prop. to flip
-3 < y < 2
15. -5
2 – x or 6x + 5
71
-7 < - x
7>x
Subtract 2 from both sides
Divide by -1 and flip sign
6x > 66
x> 11
Subtract 5 from both sides
Divide by 6
13
16. 2x + 3 < 9 or -3x – 6 < 12
2x < 6
x<3
Subtract both sides by 3
Divide both sides by 2
-3x < 18
x > -6
Add 6 to both sides
Divide by -3 and flip the sign!
x < 3 or x > -6 so… ALL REAL NUMBERS!
x< 7 or x > 11
17. Graph the answer to #13 and 15.
-3 -2 -1 0 1 2 3 4 5 6 7
5
6
7
8
9
10
11
12
Write an inequality that matches the graph.
18.
19.
-3
-3 < x < 2
x < -2
or x > 1
Properties and using properties to justify the steps of an equation.
20. Give a mathematical example for each of the following properties: Answers may vary
a. Reflexive x = x
b. Transitive If x = 2 and y =2, then x = y
c. Associative for addition (2 + 4) + 5 = 2 + (4 + 5)
d. Commutative for addition
3+7=7+3
21. Fill in the property that justifies each step.
14 = 2(x – 3)
14 = 2x – 6
Given
Distributive Property
14 + 6 = 2x – 6 + 6
Addition Property of Equality
20 = 2x + 0
Additive Inverse
20 = 2x
Additive Identity
Multiplication Property of Equality
10 = 1x
Multiplicative Inverse
10 = x
Multiplicative Identity
x = 10
Symmetric Property
22..
Solution
(3x + 9) + 1 > 13
Property
Given
3x + (9 + 1) > 13
Associative Property of Addition
3x + 10 > 13
Substitution
3x + 10 + (-10) > 13 + (-10)
Addition Property of Inequality
3x + 0 > 3
Additive Inverse
3x > 3
Additive Identity
3x >
Multiplication Property of Inequality
3
1x > 1
Multiplicative Inverse
x>1
Multiplicative Identity
23. There are three mistakes in the proof below. Find and correct the mistakes.
2(x•5 + 3) > -9
Given
2(5x + 3) > -9
Associative Property of Multiplication
commutative
10x + 6 > -9
Distributive Property
10x + 6 + (-6) > -9 + (-6)
Addition Property of Inequality
10x + 0 > -15
Inverse Property of Multiplication Addition
10x > -15
Identity Property of Addition
10x > -15 •
Multiplicative Property of Equality Inequality
1x > -
Inverse Property of Multiplication
x>-
Identity Property of Multiplication
Study tip – Go back and redo any missed questions on your old HW assignments!
Reworking the problem and arriving at the correct answer (which you hopefully wrote
down when we went over it in class) will help you prepare for the test.