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§ 2.6

§ 2.6. Solving Linear Inequalities. Linear Inequalities in One Variable, p159. Any equation in the form is called a linear inequality in one variable. The inequality symbol may be < less than > greater than ≤ less than or equal to ≥ greater than or equal to.

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§ 2.6

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  1. §2.6 Solving Linear Inequalities

  2. Linear Inequalities in One Variable, p159 Any equation in the form is called a linear inequality in one variable. The inequality symbol may be < less than > greater than ≤ less than or equal to ≥ greater than or equal to Blitzer, Introductory Algebra, 5e – Slide #2 Section 2.6

  3. Linear Inequalities in One Variable, p159 cp1 on p 160 ) [ ( ] cp2 on p 162 Blitzer, Introductory Algebra, 5e – Slide #3 Section 2.6

  4. Properties of Inequalities, p 163 • The Addition Property of Inequality If a < b, then a + c < b + c If a < b, then a – c < b – c Do cp3 on p 164 Do cp4 on p 164 Blitzer, Introductory Algebra, 5e – Slide #4 Section 2.6

  5. Properties of Inequalities, p 163 cp3 on p 164 ) cp4 on p 164 [ Blitzer, Introductory Algebra, 5e – Slide #5 Section 2.6

  6. Properties of Inequalities, p 163 • The Positive Multiplication Property of Inequality If a < b, and c is positive, then a c < b c If a < b, and c is positive then a / c < b / c Blitzer, Introductory Algebra, 5e – Slide #6 Section 2.6

  7. Properties of Inequalities, p 163 cp5a on p 165 ) Blitzer, Introductory Algebra, 5e – Slide #7 Section 2.6

  8. Properties of Inequalities, p 163 • The Negative Multiplication Property of Inequality If a < b, and c is negative, then a c < b c If a < b, and c is negative then a / c > b / c If we multiply or divide both sides of an inequality by a negative, we must change the direction of the inequality symbol. Blitzer, Introductory Algebra, 5e – Slide #8 Section 2.6

  9. Properties of Inequalities, p 163 cp5b on p 165 ( Sign changes Blitzer, Introductory Algebra, 5e – Slide #9 Section 2.6

  10. Inequalities, p 165 Blitzer, Introductory Algebra, 5e – Slide #10 Section 2.6

  11. Linear Inequalities, (similar to table 2.3 on p 161) ( ) a b [ ] a b [ ) a b ( ] a b ( a [ a ) b ] b Blitzer, Introductory Algebra, 5e – Slide #11 Section 2.6

  12. Linear Inequalities, p 165 EXAMPLE Solve the linear inequality. Then graph the solution set on a number line. SOLUTION 1) Simplify each side. Distribute 2) Collect variable terms on one side and constant terms on the other side. Add 5x to both sides Add 1 to both sides Blitzer, Introductory Algebra, 5e – Slide #12 Section 2.6

  13. Linear Inequalities CONTINUED 3) Isolate the variable and solve. Divide both sides by 8 4) Express the solution set in set-builder or interval notation and graph the set on a number line. [ -1 0 1 2 3 4 5 6 {x|x 2} = [2, ) Blitzer, Introductory Algebra, 5e – Slide #13 Section 2.6

  14. Linear Inequalities cp6 on p 166 [ Blitzer, Introductory Algebra, 5e – Slide #14 Section 2.6

  15. Linear Inequalities Example 7 on p 166 Sub 15 Sub 13x Sign changes ] Blitzer, Introductory Algebra, 5e – Slide #15 Section 2.6

  16. Linear Inequalities cp7 on p 167 Add 2 Add 3x same [ Blitzer, Introductory Algebra, 5e – Slide #16 Section 2.6

  17. Linear Inequalities cp8 on p 167 Add 8 Sub 2x [ Blitzer, Introductory Algebra, 5e – Slide #17 Section 2.6

  18. Linear Inequalities EXAMPLE Solve the linear inequality. Then graph the solution set on a number line. SOLUTION First we need to eliminate the denominators. Multiply by LCD = 10 Distribute Blitzer, Introductory Algebra, 5e – Slide #18 Section 2.6

  19. Linear Inequalities CONTINUED 1 2 1 1 1 1 1) Simplify each side. Because each side is already simplified, we can skip this step. 2) Collect variable terms on one side and constant terms on the other side. Add x to both sides Subtract 10 from both sides Blitzer, Introductory Algebra, 5e – Slide #19 Section 2.6

  20. Linear Inequalities CONTINUED 3) Isolate the variable and solve. Divide both sides by 4 4) Express the solution set in set-builder or interval notation and graph the set on a number line. [ -6 -5 -4 -3 -2 -1 0 1 2 {x|x -2} = [-2, ) Blitzer, Introductory Algebra, 5e – Slide #20 Section 2.6

  21. Properties of Inequalities • Do problems 62, 64, 68, and 72 Blitzer, Introductory Algebra, 5e – Slide #21 Section 2.6

  22. DONE

  23. Linear Inequalities (Extra) EXAMPLE Graph the solution of each inequality on a number line and then express the solutions in set-builder notation and interval notation. SOLUTION (a) The solution to x > -4 is all real numbers that are greater than -4. They are graphed on a number line by shading all points to the right of -4. The parenthesis at -4 indicates that -4 is not a solution, but numbers such as -3.9999 and -3.3 are. The arrow shows that the graph extends indefinitely to the right. {x|x > -4} = (-4, ) ( -4 Blitzer, Introductory Algebra, 5e – Slide #23 Section 2.6

  24. Linear Inequalities(Extra) CONTINUED (b) The solution to is all real numbers that are greater than -2 and less than 7. They are graphed on a number line by shading all points that are to the right of -2 and to the left of 7. The bracket at -2 indicates that -2 is part of the solution. The parenthesis at 7 indicates that 7 is not part of the solution. {x|-2 x < 7} = [-2,7) ) [ -2 7 Blitzer, Introductory Algebra, 5e – Slide #24 Section 2.6

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