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1-4 Solving Inequalities

1-4 Solving Inequalities. M11.D.2.1.1: Solve compound inequalities and/or graph their solution sets on a number line. Objectives. Solving and Graphing Inequalities Compound Inequalities. Key Concepts. Transitive Property If a ≤ b and b ≤ c, then a ≤ c Ex. if 2 ≤ 5 and 5 ≤ 11, then 2 ≤ 11

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1-4 Solving Inequalities

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  1. 1-4 Solving Inequalities M11.D.2.1.1: Solve compound inequalities and/or graph their solution sets on a number line

  2. Objectives Solving and Graphing Inequalities Compound Inequalities

  3. Key Concepts • Transitive Property If a ≤ b and b ≤ c, then a ≤ c Ex. if 2 ≤ 5 and 5 ≤ 11, then 2 ≤ 11 • Addition Property If a ≤ b, then a + c ≤ b + c If 2 ≤ 5, then 2 + 10 ≤ 5 + 10 • Subtraction Property If a ≤ b, then a - c ≤ b – c If 10 ≤ 15, then 10 – 2 ≤ 15 – 2

  4. Key Concepts • Multiplication Property If a ≤ b and c > 0, then ac ≤ bc If 3 ≤ 5 and 2 > 0, then 3(2) ≤ 5(2) If a ≤ b and c < 0, then ac ≥bc If 3 ≤ 5 and -2 < 0, then 3(-2) ≥ 5(-2) If you multiply by a negative number, the sign switches

  5. Key Concepts • Division Property If a ≤ b and c > 0, then ≤ If 3 ≤ 5 and 2 > 0, then ≤ If a ≤ b and c < 0, then ≥ If 3 ≤ 5 and -2 < 0, then ≥ If you multiply by a negative number, the sign switches

  6. How to Graph Inequalities Graph x > -3 • Step One – Mark the number with a circle • If the inequality is > or <, don’t fill in the circle (Open) • If the inequality is ≥ or ≤, fill in the circle (Closed) • Step Two – Shade the line for all true values of x • If x is greater than a number, then shade to the right. • If x is less than a number, then shade to the left.

  7. Another Example Graph x ≤ 2

  8. Solving and Graphing Inequalities Solve –2x < 3(x – 5). Graph the solution. –2x < 3(x – 5) –2x < 3x – 15 Distributive Property –5x < –15 Subtract 3x from both sides. x > 3 Divide each side by –5 and reverse the inequality.

  9. > > > > > – – – – – 7x 7(2 + x) 7x 14 + 7x Distributive Property 0 14Subtract 7x from both sides. The last inequality is always false, so 7x 7(2 + x) is always false. It has no solution. No Solutions or All Real Numbers as Solutions Solve 7x 7(2 + x). Graph the solution.

  10. > > > > – – – – Relate: $2000 + 4% of sales $5000 Define: Let x = sales (in dollars). Write: 2000 + 0.04x 5000 0.04x 3000 Subtract 2000 from each side. x 75,000 Divide each side by 0.04. Ex 1 & 2 as a Word Problem A real estate agent earns a salary of $2000 per month plus 4% of the sales. What must the sales be if the salesperson is to have a monthly income of at least $5000? The sales must be greater than or equal to $75,000.

  11. Vocabulary A compound inequality is a pair of inequalities joined by and or or. Ex. -1 < x and x ≤ 3, can also be written as -1 < x ≤ 3

  12. < < < < < – – – – – 2x – 1 3x and x > 4x – 9 –1 x 9 > 3x –1 x and 3 > x This compound inequality can be written as –1 x < 3. Compound Inequality Containing And Graph the solution of 2x – 1 3x and x > 4x – 9.

  13. 3x < –12 –2x < 4 Compound Inequality Containing Or Graph the solution of 3x + 9 < –3 or –2x + 1 < 5. 3x + 9 < –3 or –2x + 1 < 5 x < –4 or x > –2

  14. < < < < < < < < – – – – – – – – > > – – Relate: minimum length final length maximum length Define:  Let x = number of centimeters to remove. Write: 17 – 0.15 18 – x 17 + 0.15 16.85 18 – x 17.15 Simplify. –1.15 – x –0.85 Subtract 18. 1.15 x 0.85 Multiply by –1. Ex. 4 & 5 as a Word Problem A strip of wood is to be 17 cm long with a tolerance of ± 0.15 cm. How much should be trimmed from a strip 18 cm long to allow it to meet specifications? At least 0.85 cm and no more than 1.15 cm should be trimmed off to meet specifications.

  15. Homework Take out a piece of new lined paper (3 hole punched) Please put your name on the top left line and the information below on the top right. Pg 29 & 30 #1, 2, 14, 15, 18, 19, 22, 23, 26, 27

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