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Mastering Linear Inequalities: Solving, Graphing, and Real-World Applications

This resource covers the fundamentals of linear inequalities in algebra. Learn how to solve for variables, understand inequality symbols (>, <, ≥, ≤), and graph solutions effectively. The guide provides step-by-step examples, warm-up exercises, and practice problems to hone your skills. Additionally, it delves into the importance of checking solutions by substituting values back into original inequalities. With plenty of opportunities for guided practice, you’ll solidify your understanding of how to handle linear inequalities and prepare for more advanced concepts.

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Mastering Linear Inequalities: Solving, Graphing, and Real-World Applications

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  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.

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