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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting AF4.0 Students solve simple linear equations and inequalities over the rational numbers. California Standards

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting When you add or subtract the same number on both sides of an inequality, the resulting statement will still be true. –2 < 5 +7 5 < 12 You can find solution sets of inequalities the same way you find solutions of equations, by isolating the variable.

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting Solve and graph the inequality. Additional Example 1A: Solving Inequalities by Adding or Subtracting x + 3 > –5 –3 x > –8 Since 3 is added x, subtract 3 from both sides. –9 8 –7 6 5 4 3 2 1 0 1 2 3 4 5

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting Additional Example 1A Continued Check x + 3 > –5 –4 + 3 > –5 ? –1 > –5 ? Substitute –4 for x. According to the graph –4 should be a solution and –9 should not be a solution. So –4 is a solution. x + 3 > –5 –9 + 3 > –5 ? –6 > –5 ? Substitute –9 for x. So –9 is not a solution.

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting m – 4 ≥ –2 + 4 m ≥ 2 Since 4 is subtracted from m, add 4 to both sides. –1 0 1 2 3 4 5 6 7 8 9 10 11 12 Solve and graph the inequality. Additional Example 1B: Solving Inequalities by Adding or Subtracting

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting r + 3 ≤ –3 – 3 –3 r ≤ –6 Since 3 is added to r, subtract 3 from both sides. Additional Example 1C: Solving Inequalities by Adding or Subtracting –9 8 –7 6 5 4 3 2 1 0 1 2 3 4 5 Solve and graph the inequality.

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting 5 > n + 1 Since 1¼ is added to n, subtract 1¼ from both sides. Additional Example 1D: Solving Inequalities by Adding or Subtracting Solve and graph the inequality. 3434 1414 5 > n + 1 3434 1414 – 1 1414 1414 4 > n 1212 –7 6 –5 4 3 2 1 0 1 2 3 4 5 6 7

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting Solve and graph the inequality. Check It Out! Example 1A x + 4 > –2 –4 x > –6 Since 4 is added x, subtract 4 from both sides. –9 8 –7 6 5 4 3 2 1 0 1 2 3 4 5

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting Check It Out! Example 1A Continued Check x + 4 > –2 2 + 4 > –2 ? 6 > –2 ? Substitute 2 for x. According to the graph 2 should be a solution and –8 should not be a solution. So 2 is a solution. x + 4 > –2 –8 + 4 > –2 ? –4 > –2 ? Substitute –8 for x. So –8 is not a solution.

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting w – 8 ≥ –3 + 8 w ≥ 5 Since 8 is subtracted from w, add 8 to both sides. –1 0 1 2 3 4 5 6 7 8 9 10 11 12 Solve and graph the inequality. Check It Out! Example 1B

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting c + 6 ≤ –1 – 6 c ≤ –7 Since 6 is added to c, subtract 6 from both sides. Check It Out! Example 1C –9 8 –7 6 5 4 3 2 1 0 1 2 3 4 5 Solve and graph the inequality.

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting 3 > n + 1 Since 1 is added to n, subtract 1 from both sides. Check It Out! Example 1D Solve and graph the inequality. 2323 1313 3 > n + 1 2323 1313 – 1 1313 1313 2 > n 1313 –7 6 –5 4 3 2 1 0 1 2 3 4 5 6 7 1313 1313

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting While training for a race, Ann’s goal is to run at least 3.5 miles each day. She has already run 1.8 miles today. Write and solve an inequality to find out how many more miles she must run today. Additional Example 2: Sports Application Let m = the number of additional miles. 1.8 + m ≥ 3.5 –1.8 m ≥ 1.7 Ann should run at least 1.7 more miles. Since 1.8 is added to m, subtract 1.8 from both sides. 1.8 miles plus additional miles is at least 3.5 miles. 1.8 + m ≥ 3.5

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting 2 is greater than 1.7. Substitute 2 for m. Check Additional Example 2 Continued 1.8 + m ≥ 3.5 1.8 + 2 ≥ 3.5 ? 3.8 ≥ 3.5 ? x 1 is less than 1.7. Substitute 1 for m. 1.8 + m ≥ 3.5 1.8 + 1 ≥ 3.5 ? 2.8 ≥ 3.5 ?

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting Tim’s company produces recycled paper. They produce 60.5 lb of paper each day. They have already produced at least 20.2 lb today. Write and solve an inequality to find out how many more pounds Tim’s company must produce. Check It Out! Example 2 Let p = the number of additional pounds of paper. 20.2 + p ≥ 60.5 –20.2 –20.2 p ≥ 40.3 Tim’s company should produce at least 40.3 lb more of paper. Since 20.2 is added to p, subtract 20.2 from both sides. 20.2 lbs plus additional pounds is at least 60.5 lb. 20.2 + p ≥ 60.5

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Evaluating Algebraic Expressions 3-6 Solving Inequalities by Adding or Subtracting 41 is greater than 40.3. Substitute 41 for p. Check Check It Out! Example 2 Continued 20.2 + p ≥ 60.5 20.2 + 41 ≥ 60.5 ? 61.2 ≥ 60.5 ? x 40 is less than 40.3. Substitute 40 for p. 20.2 + p ≥ 60.5 20.2 + 40 ≥ 60.5 ? 60.2 ≥ 60.5 ?

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An equation is a mathematical statement that two expressions are equivalent. The solution set of an equation is the value or values of the variable that.

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