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Section 9.2 More Equations and Inequalities

In section 9.1, we learned how to solve absolute value equations. In this section, we will learn how to solve absolute value inequalities. Some examples of absolute value inequalities are 9.2 Solving Absolute Value Inequalities

Solve Absolute Value Inequalities Containing < or What does it mean to solve. It means to find the set of all real numbers whose distance from zero is 2 units or less. 2 is 2 units from is 2 units from 0. Any number between 2 and -2 is less than 2 units from zero. For example, if x = 1,. If x = -1,. We can represent the solution set on a number line as We can write the solution set in interval notation as [-2, 2].

Example 5 Solution We must find the set of all real numbers whose distance from zero is less than 3. We can do this by solving the three-part inequality. We can represent this on a number line as: We can write the solution set in interval notation as (-3,3). Any number between -3 and 3 will satisfy the inequality

Example 6 Solution We must find the set of all real numbers, a, so that a + 4 is less than or equal to 10 Units from zero. To solve, we must solve the three-part inequality. Subtract 4. The number line representation is In interval notation, we write the solution as [-14, 6]. Any number between -14 and 6 Will satisfy the inequality

Solve Absolute Value Inequalities Containing > or What does it mean to solve. It means to find the set of all real numbers whose distance from zero is 2 units or more. 2 is 2 units from is 2 units from 0. Any number greater than 2 or less than -2 is more than 2 units from zero. Any number greater than 2 or less than -2 is more than 2 units from zero. For example, if x = 3,. If x = -4,. We can represent the solution set on a number line as The solution set consists of two separate regions, so we can write a compound Inequality using or. In interval notation, we write

Example 7 Solution We must find the set of all real numbers whose distance from zero is greater than 4. The solution is the compound inequality. On the number line, we can represent the solution set as: The solution set consists of two separate regions, so we can write a compound Inequality using or. In interval notation, we write

Example 8 Solution Begin by getting the absolute value on a side by itself. Subtract 12. Rewrite as a compound Inequality. Add 3. The graph of the solution set is: The interval notation is

Solve Special Cases of Absolute Value Inequalities Example 9 Solution a)Look carefully at this inequality,. It says that the absolute value of a quantity, h + 2 is less than a negative number. Since the absolute value of a quantity is always zero or positive, this inequality has no solution. The solution set is b) says that the absolute value of a quantity, 4q + 9, is greater than or equal to zero. An absolute value is always greater than or equal to zero, so any value of q will make this inequality true. The solution set consists of all real numbers, which we can write in interval notation as. c) Begin by isolating the absolute value. Subtract 6. The absolute value of a quantity can never be less than zero but it can equal zero. To solve this, we must solve 3t - 5 =0. Subtract 5. Divide 3 The solution set is