Graphing Linear Inequalities

1. Graphing Linear Inequalities 6.1 & 6.2

2. x 1. ≤ 7; 20 3 f 4.Solve the equation of + 8 = 7. 4 Warm-Up Check whether the given number is a solution of the Inequality. ANSWER yes 2. 4m > –23; –6 ANSWER no 3.Is 4a solution of the inequality 2a – 1.5 ≥ 6? ANSWER yes ANSWER –4 Add. 5) 6)

3. What’s the Difference? • 4x + 2 = 3y • 4x + 2 < 3y

4. Steps to Solving Linear Inequalities • Solve for the variable. • If you multiple or divide by a negative number, then you change the sign of the inequality • < changes to > • > changes to < • Example: -2x < 5 • Otherwise, solve as you normally would (if their was an equal sign).

5. b. a. Graph the Linear Inequality • Graph it on a number line. •  The number is not a solution (> or <) •  The number is a solution (≤ or ≥) • x > -6.5 • x ≤ 4

6. Solve x – 5 > – 3.5.Graph your solution. ANSWER The solutions are all real numbers greater than 1.5. Check by substituting a number greater than 1.5 for xin the original inequality. x – 5 > – 3.5 Write original inequality. x – 5 +5> – 3.5 +5 Add 5 to each side. x > 1.5 Simplify.

7. The solutions are all real numbers less than 12. Check by substituting a number less than 12 for xin the original inequality. ANSWER GUIDED PRACTICE Solve the inequality. Graph your solution. 4. x – 9 ≤ 3 SOLUTION x – 9 ≤ 3 Write original inequality. x – 9 +9 ≤ 3+9 Add 9 to each side. x ≤ 12 Simplify.

8. ANSWER The solutions are all real numbers less than 4.2. Check by substituting a number less than 4.2 for xin the original inequality. GUIDED PRACTICE 5. p – 9.2 < – 5 SOLUTION p – 9.2 < – 5 Write original inequality. p – 9.2 +9.2 < – 5+9.2 Add 9.2 to each side. x < 4.2 Simplify.

9. ANSWER 1 1 1 1 1 1 2 2 2 2 2 2 The solutions are all real numbers less than . Check by substituting a number less than for xin the original inequality. 6. – 1 ≥ m – – 1 ≥ m – 1 1 – 1+≥ m+ Add to each side. 2 2 1 ≥ m – – – 2 GUIDED PRACTICE Write original inequality. Simplify.

10. Solve 9x+ 7. Graph your solution. ANSWER You can rewrite2 ≥ x asx ≥ 2. The solutions are all real numbers less than or equal to 2. > EXAMPLE 4 9≥ x+ 7 Write original inequality. 9– 7 ≥x+ 7 – 7 Subtract 7 from each side. 2 ≥ x Simplify.

11. x x x . Graph your solution. Solve 4 4 4 5. 5 4 < 4 5 ANSWER The solutions are all real numbers less than20.Check by substituting a number less than20 in the original inequality. < < EXAMPLE 1 Write original inequality. Multiply each side by4. x < 20 Simplify.

12. x x x 3 3 3 8. 8 1.> 8 3 3 > > Solve the inequality. Graph your solution. SOLUTION Write original inequality. Multiply each side by8. x > 24 Simplify. ANSWER The solutions are all real numbers are greater than 24. Check by substituting a number greater than 24 in the original inequality.

13. m m m 8 8 8 2.≤ – 2 ≤ – 2 ≤ – 2 8 3 GUIDED PRACTICE SOLUTION Write original inequality. Multiply each side by8. m < – 16 Simplify. ANSWER The solutions are all real numbers less than are equalto– 16. Check by substituting a number less than – 16 in the original inequality.

14. y y y 3.≥ – 4 2.5 2.5 2.5 ≥ – 4 ≥ – 4 8 2.5 GUIDED PRACTICE SOLUTION Write original inequality. Multiply each side by2.5. y ≥ – 10 Simplify. The solutions are all real numbers less than are equalto– 10. Check by substituting a number greater than – 10 in the original inequality. ANSWER

15. x x x <7.Graph your solution. Solve –6 –6 –6 –6 >–6 7 <7. ANSWER The solutions are all real numbers greater than – 42. Check by substituting a number greater than – 42 in the original inequality. EXAMPLE 2 Write original inequality. Multiply each side by –6. Reverse inequality symbol. x > – 42 Simplify.

16. < 7 < 7 ? x 0 – 6 – 6 EXAMPLE 2 CHECK Write original inequality. Substitute 0 for x. 0 < 7 Solution checks

17. Solve –3x >24. < –3x 24 –3 –3 EXAMPLE 3 –3x >24. Write original inequality. Divide each side by –3. Reverse inequality symbol. x < – 8 Simplify.

18. 4.> 12 x x x – 4 – 4 – 4 > 12 – 4 < – 4 12 ANSWER The solutions are all real numbers greater than are equalto– 48. Check by substituting a number greater than – 48 in the original inequality. SOLUTION Write original inequality. Multiply each side by–4. x <– 48 Simplify.

19. 5.< 1.6 m m m – 7 – 7 – 7 < 1.6 – 7 < – 7 1.6 SOLUTION Write original inequality. Multiply each side by–7. m < – 11.2 Simplify.

20. 5v 45 ≥ 5 5 ANSWER The solutions are all real numbers greater than are equalto 9. Check by substituting a number greater than 9 in the original inequality. 6. 5v ≥ 45 SOLUTION 5v ≥ 45 Write original inequality. Divide both the side by 5. v ≥ 9 Simplify.

21. 24 –6n > 6 6 ANSWER The solutions are all real numbers greater than are equalto –4. Check by substituting a number greater than –4 in the original inequality. GUIDED PRACTICE 7. –6n < 24 SOLUTION –6n < 24 Write original inequality. Divide both the side by 6. n > –4 Simplify.

22. Solve 3x– 7 < 8.Graph your solution. ANSWER The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality. EXAMPLE 1 Solve a two-step inequality 3x– 7 < 8 Write original inequality. 3x< 15 Add 7 to each side. x< 5 Divide each side by 3.

23. ? 3(0) – 7 < 8 EXAMPLE 1 Solve a two-step inequality CHECK 3x –7 < 8 Write original inequality. Substitute 0 for x. –7 < 8 Solution checks.

24. Solve –0.6(x– 5)<15 – –0.6(x–5)<15 – –0.6x+ 3 <15 – –0.6x<12 – – Divide each side by 0.6. Reverse inequality symbol. x > –20 – EXAMPLE 2 Solve a multi-step inequality Write original inequality. Distributive property Subtract 3 from each side.

25. < < < < 1. 2x– 5 23. – – – – 2x– 5 23 2x 28 x 14 ANSWER The solutions are all real numbers less than equal to 14. Check by substituting a number less than 14. Solve the inequality. Graph your solution. Write original inequality. Add 5 to both side. Divide each side by 2.

26. < < < > 2. – 6y+5 –16. – – – – – 6y+5 –16 –16 –5 y –6 y 3.5 ANSWER The solutions are all real numbers less than or equal to 3.5. Check by substituting a number less than 3.5. Solve the inequality. Graph your solution. Write original inequality. Divide the equation by 6. Simplify.

27. –1 –1 ( p –12) > –2 3. 4 4 ( p –12) > 12 ANSWER The solutions are all real numbers greater than 20. Check by substituting a number greater than 20. Solve the inequality.graph your solution Write original inequality. – p + 12 > –8 Distributive property. – p > –8 – 12 p < 20

28. ANSWER There are no solutions because 25 < 17is false. Identify the number of solutions of an inequality Solve the inequality, if possible. Graph your solution. 5. 5(m + 5) < 5m + 17 SOLUTION 5(m + 5) < 5m + 17 Write original inequality. 5m + 25 < 5m + 17 Distributive property 25 < 17 Subtract 5mfrom each side.

29. 6. < < < < < 1– 8s–4(2s –1). – – – – – 1– 8s–4(2s –1) 1– 8s8s + 4 1 4 ANSWER All real numbers are solutions because is true. 1 4 Identify the number of solutions of an inequality Solvethe inequality,if possible. Graph your solution. SOLUTION Write original inequality. Distributive property Subtract 8sfrom each side.