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4.1 Solving Linear Inequalities 11/2/2012

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You have learned how to solve equations with 1 variable. Ex. x + 3 = 7 -3 -3 x = 4 Ex. x - 5 = 2 +5 +5 x = 7 Ex. 3x = 12 3 3 x = 4 Ex. y = 6 2 y = 12 ·2 2·

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To solve inequality in 1 variable All the rules apply for solving equations in 1 variable except when dividing or multiplying both sides by a negative number. Rule: When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

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Example 1 Inequality with a Variable on One Side Solve the inequality. a.b. > x 4 – 6 –– ≥ 5y 213 – + SOLUTION a. > x 4 – 6 – Write original inequality. > x2 – Add 4 to each side. ANSWER The solution is all real numbers greater than 2. – + 4

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Example 1 Inequality with a Variable on One Side b. – ≥ 5y 213 – + Write original inequality. – ≥ 5y5y15 – Subtract 2 from each side. ANSWER The solution is all real numbers less than or equal to 3. Divide each side by 5 and reverse the inequality. – y ≤ 3 -2 -5

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Graphing the solution of an inequality in 1 variable. For > or <, doughnut ( ) For ≤ or ≥, solid circle ( ) Then shade the side of the number line according to which way the inequality symbol is pointing when the variable is on the left side Ex. x > -1 Ex. x ≤ 3

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Example 2 Inequality with a Variable on Both Sides Solve. Graph the solution. Add 2x to each side. < 7 2x – 1 < 7 4x1 2x –– SOLUTION Write original inequality. < 7 4x1 2x –– < 2x2x – 6 – Subtract 7 from each side. > x3 Divide each side by 2 and reverse the inequality. – +2x -2

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Example 2 Inequality with a Variable on Both Sides ANSWER The solution is all real numbers greater than 3. The graph is shown at the bottom. 234511 – 0

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Checkpoint Solve the inequality. Then graph your solution. Solve an Inequality <x 38 + 1. 5 ≤ 4 x – 2. ANSWER < x5 2103456 ≥x 1 – 02 2 –

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Checkpoint Solve the inequality. Then graph your solution. – x > 2x 3 4. Solve an Inequality 2 > 2x 1 – 3. 3 x > ANSWER 2103456 x > 2 3 12 0

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Example 3 Use a Simple Inequality Amusement Park Admission to an amusement park costs $9 and each ride ticket costs $1.50. The total amount A in dollars spent is given by where t is the number of ride tickets. Use an inequality to describe the number of ride tickets you can buy if you have at most $45 to spend at the park. = A1.5t + 9 SOLUTION The most money you can spend is $45. A45 ≤ Substitute for A. 45 ≤ 1.5t + 9 + 9 Subtract 9 from each side. 36 ≤ 1.5t Divide each side by 1.5. 24 ≤ t ANSWER You can buy up to 24 ride tickets.

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COMPOUND INEQUALITY 2 simple inequalities joined by the word “and” or the word “or”. Ex: AND All real numbers greater than or equal to -2 AND less than 1. Ex: OR All real numbers less than -1 OR greater than or equal to 2 -2≤ x < 1 x < -1 or x ≥ 2

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Example 4 Solve an “Or” Compound Inequality Solve or 3x3x + 28< 2x2x9 3.3. – > SOLUTION Solve each part separately. FIRST INEQUALITYSECOND INEQUALITY 3x3x + 28<2x2x93 – > Write first inequality. Write second inequality. 3x3x6<2x2x12 > Subtract 2 from each side. Add 9 to each side. x2<x6 > Divide each side by 3. Divide each side by 2.

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Checkpoint Solve the inequality. Then graph your solution. Solve Compound Inequalities 5. 4x< + 57< 6. 3x3x ≤ + 81 – 8 ≤ ANSWER x2<1< – 2 – 20 x ≤ 3 – 0 ≤ 4 – 2 – 0..

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Checkpoint Solve the inequality. Then graph your solution. Solve Compound Inequalities 7. x + 3 4 or ≤ x6 –≥ 1 – ANSWER x 1 or ≤ x5 ≥ 420.. 8. 2x2x60< –– x 4 or – > ANSWER x 4 or – <x3 – > 0 6 – 4 – 2 –

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Homework 4.1 p.175 #14-22 even, 23-28all, 30-38even, 40-44all, 50-56even

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