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Absolute Value Equations and Inequalities Page 1 of 3 Absolute Value Equations and Inequalities The absolute value of a number is simply “how big” the number is, without a + or - sign in front of it. The technical definition is given on page 140 of the textbook, but the meaning is the common one … e.g. |4| = 4 and |-4| = 4. Since there are always two numbers with the same absolute value, solving absolute value equations requires a little care. We use the same techniques and procedures used to solve “normal” equations, but we have to check out two possible solutions. We’ll look at three cases here: (1) solving and equation for the absolute value of a variable, (2) solving an equation for the absolute value of a function of the variable, and (3) solving inequalities with absolute values. The difference here will be that we’ll have to add a step, since the absolute value can arise from two different numbers. 1. Solving equations for absolute value of variable: This is straight forward: we solve the equation as we would any other equation, and then we consider the two possibilities. Let’s look at an example. |3x| + 5 = 14 We want to isolate the variable and then look at the two possible answers. |3x| + 5 - 5 = 14 - 5 |3x| = 9 One solution: 3x = +9 x=3 The other solution: 3x = -9 x = -3 Then our solutions are 3 and -3, and we can write the solution set as {3, - 3}. 2. Solving equations for the absolute value of a function of the variable: Actually, we just did this in the previous example: “3x” is not the variable itself, but a function of the variable (three times the variable). Let’s look at an example: |4z - 2| + 8 = 46 First we isolate the absolute value term. |3z -2| + 8 - 8 = 46 - 8 |3z - 2| = 38 Now we look at the two possibilities: http://66.20.58.22/courses/1/SUCCESS/content/_28527_1/Absol_value_eq_ineq.htm 6/23/2003 Absolute Value Equations and Inequalities Page 2 of 3 First the “positive” case: 3z - 2 = 38 3z - 2 + 2 = 38 + 2 3z = 40 z = 10 And the “negative” case: 3z - 2 = -38 3z - 2 + 2 = -38 + 2 = - 36 3z = -36 z = -9 So our solution set is {10, -9}. 3. Absolute values on both sides: This looks a little daunting at first, but the principle is the same … we consider the “positive” case (make the right side positive) and the “negative” case (make the right side negative. Let's look at an example: |9y + 1| = |6y + 4| When we consider a case, we discard the absolute value signs. Let’s start with the right side positive: 9y + 1 = +(6y + 4) 9y + 1 = 6y + 4 9y + 1 - 1 = 6y + 4 - 1 9y = 6y + 3 9y - 6y = 6y - 6y + 3 3y = 3 y=1 That’s one solution. Now let’s make the right side negative, being careful to distribute that negative sign: 9y + 1 = -(6y + 4) 9y + 1 = -6y - 4 9y + 1 - 1 = -6y - 4 - 1 9y = -6y - 5 9y + 6y = -6y + 6y - 5 15y = -5 y = -5/15 = -1/3 y = -1/3 The solution is then {1, - 1/3}. Solving inequalities: There is a special way to handle inequalities with absolute values … we write the absolute value as a compound inequality and solve that. Let's look at an example: http://66.20.58.22/courses/1/SUCCESS/content/_28527_1/Absol_value_eq_ineq.htm 6/23/2003 Absolute Value Equations and Inequalities Page 3 of 3 |3x| + 5 < 17 First, we need to isolate the absolute value expression: |3x| + 5 - 5 < 17 - 5 |3x| < 12 Now we use the positive case … 3x < 12 … and the negative case … 3x > -12, or -12 < 3x … to write a compound inequality. -12 < 3x < 12 Solving this, we get -4 < x < 4 and the solution set is (4, 4). http://66.20.58.22/courses/1/SUCCESS/content/_28527_1/Absol_value_eq_ineq.htm 6/23/2003