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Section 2.5

Section 2.5. Linear Inequalities. Page 136. Solutions and Number Line Graphs. A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols <, ≤, >, or ≥. x > 5, 3 x + 4 < 0, 1 – y ≥ 9

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Section 2.5

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  1. Section 2.5 • Linear Inequalities

  2. Page 136 Solutions and Number Line Graphs • A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols <, ≤, >, or ≥. • x > 5, 3x + 4 < 0, 1 – y ≥ 9 • A solution to an inequality is a value of the variable that makes the statement true. The set of all solutions is called the solution set.

  3. Example Page 137 • Use a number line to graph the solution set to each inequality. • a. • b. • c. • d.

  4. Linear Inequalities in One Variable Page 137 ) [ ( ]

  5. Page 137 Interval Notation Each number line graphed on the previous slide represents an interval of real numbers that corresponds to the solution set to an inequality. Brackets and parentheses can be used to represent the interval. For example:

  6. Example Interval Notation Page 137 • Write the solution set to each inequality in interval notation. • a. b. • Solution • a. b. • More examples

  7. Example Checking a Solution Page 138 • Determine whether the given value of x is a solution to the inequality. • Solution

  8. The Addition Property of Inequalities Page 139

  9. Example Page 139 • Solve each inequality. Then graph the solution set. • a. x – 2 > 3 b. 4 + 2x ≤ 6 + x • Solution a. x – 2 > 3 b. 4 + 2x ≤ 6 + x x – 2 + 2 > 3 + 2 4 + 2x– x≤ 6 + x– x 4 + x ≤ 6 x > 5 4 – 4 + x ≤ 6 – 4 x ≤ 2

  10. Properties of Inequalities Page 140 ) [

  11. The Multiplication Property of Inequalities Page 141

  12. Example Page 141 • Solve each inequality. Then graph the solution set. • a. 4x > 12 b. • Solution • a. 4x > 12 b.

  13. Example Page 142 • Solve each inequality. Write the solution set in set-builder notation. • a. 4x – 8 > 12 b. • Solution • a. 4x – 8 > 12 b.

  14. Properties of Inequalities Page 142 ) ( Sign changes

  15. Linear Inequalities Page 142 Add 2 Subtract 6 Add 3x Subtract 5x Sign changes same [

  16. Linear Inequalities Page 142 Add 8 Sub 2x [

  17. Page 143 Applications • To solve applications involving inequalities, we often have to translate words to mathematical statements.

  18. Example Page 143 • Translate each phrase to an inequality. Let the variable be x. • a. A number that is more than 25. • b. A height that is at least 42 inches. x > 25 x ≥ 42

  19. Example Page 144 • For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. • a. Write a formula that gives the cost C of producing x cases of snacks. • b. Write a formula that gives the revenue R from selling x cases of snacks. C = 135x + 175,000 R = 250x

  20. Example (cont) Page 144 • For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. • c. Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x cases of snacks. P = R – C =250x– (135x + 175,000) = 115x – 175,000

  21. Example (cont) Page 144 • For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. • d. How many cases need to be sold to yield a positive profit? 115x – 175,000 > 0 115x > 175,000 x > 1521.74 Must sell at least 1522 cases.

  22. DONE

  23. Objectives • Solutions and Number Line Graphs • The Addition Property of Inequalities • The Multiplication Property of Inequalities • Applications

  24. EXAMPLE Graphing inequalities on a number line Use a number line to graph the solution set to each inequality. a. b. c. d.

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