Section 2.7 Absolute Value Inequalities 125 SECTION 2.7 Absolute Value Inequalities During the 2008 presidential election, one poll showed Obama leading McCain by 47% to 44%. This poll had a margin of error of 3%. According to the poll, what percent of votes could Obama actually get? To answer this question, we must solve the inequality up - 47u ≤ 3, where p is equal to the percent of people that actually voted for Obama. (Source: www.foxnews.com) In this section, we will learn how to solve inequalities involving absolute values. Absolute Value Inequalities with < or ≤ When we solve an absolute value equation, we are looking for values that have a specific distance from 0 on the number line. But when we solve absolute value inequalities, we are looking for numbers whose distance from 0 may be less than or greater than a given number. Suppose we want to solve uxu ≤ 2. This means that we need to find all real numbers whose distance from zero is less than or equal to 2. Consider the following number line. x –5 –4 –3 –2 –1 0 1 2 3 4 5 |x| 5 4 3 2 1 0 1 2 3 4 5 So, we see from the number line that numbers between, and including, -2 and 2 have an absolute value less than or equal to 2. So, the solution set is -2, 2. This leads to a property that will enable us to solve absolute value inequalities of the form uXu < k or uXu ≤ k. Property: Property 1 for Absolute Value Inequalities Let k be a positive real number. If uXu < k, then X < k and X > -k. If uXu ≤ k, then X ≤ k and X ≥ -k. Note: These compound inequalities can also be written as -k < X < k and -k ≤ X ≤ k Procedure: Solving an Inequality of the Form uXu < k or uXu ≤ k Step 1: Isolate the absolute value expression on one side of the inequality. Step 2: Apply property 1 to remove the absolute value sign. Step 3: Solve the resulting compound inequality. Step 4: Graph the solution set and write the answer in interval notation. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Solve absolute value inequalities involving < or ≤. 2. Solve absolute value inequalities involving > or ≥. 3. Solve special cases of absolute value inequalities. 4. Solve applications of absolute value inequalities. 5. Solve absolute value inequalities using test points. 6. Troubleshoot common errors. Objective 1 ▶ Solve absolute value inequalities involving < or ≤.
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