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Name: ________________________________________ 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.