1 / 24

§ 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.

mostyn
Download Presentation

§ 2.6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related