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**Absolute-Value Equations and Inequalities**

9.3 Absolute-Value Equations and Inequalities Equations with Absolute Value Inequalities with Absolute Value

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**Warm Up Solve and graph each compound inequality.**

1. -3x – 4 < -16 and 2x - 5 < 15 2. 4x + 2 > 10 or -2x + 6 > 16 3. 5 < -3x+2 < 8 4. 2x < -2 and x + 3 > 6 5. 5x > 15 or 6x - 4 < 32

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**Definition of absolute value.**

The absolute value of a number is its distance from 0. Since absolute value is distance it is never negative. Definition of absolute value.

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**Find each absolute value.**

|-3| |0| |4|

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**Example Solution a) |x| = 6; b) |x| = 0; c) |x| = –2**

a) |x| = 6 means that the distance from x to 0 is 6. Only two numbers meet that requirement. Thus the solution set is {–6, 6}. b) |x| = 0 means that the distance from x to 0 is 0. The only number that satisfies this is zero itself. Thus the solution set is {0}. c) Since distance is always nonnegative, |x| = –2 has no solution. Thus the solution set is

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**The Absolute-Value Principle for Equations**

This brings us to… The Absolute-Value Principle for Equations If |x| = p, then x = p or x = -p. Note: The equation |x| = 0 is equivalent to the equation x = 0 and the equation |x| = –p has no solution.

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**Example: a) |2x +1| = 5; b) |3 – 4x| = –10**

Solution a) We use the absolute-value principle, knowing that 2x + 1 must be either 5 or –5: |x| = p |2x +1| = 5 Substituting 2x +1 = –5 or 2x +1 = 5 2x = –6 or x = 4 x = –3 or x = 2 The solution set is {–3, 2}. The check is left for the student.

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**(continued) Solution b) |3 – 4x| = –10**

The absolute-value principle reminds us that absolute value is always nonnegative The equation |3 – 4x| = –10 has no solution. The solution set is To apply the absolute value principle we must make sure the absolute value expression is ISOLATED.

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**Given that f (x) = 3|x + 5| – 4, find all x for which f (x) = 11**

Solution Since we are looking for f(x) = 11, we substitute: f(x) = 11 Replacing f (x) with 3|x + 5| − 4 3|x+5| – 4 = 11 3|x + 5| = 15 |x + 5| = 5 x + 5 = –5 or x + 5 = 5 x = –10 or x = 0 The solution set is {–10, 0}. The check is left for the student.

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**When more than one absolute value expression appears…**

Sometimes an equation has two absolute-value expressions like |x+1| = |2x|. This means that x+1 and 2x are the same distance from zero. Since they are the same distance from zero, they are the same number or they are opposites. So solve x+1 = 2x and x+1 = -2x separately to get your answers. As always check by substitution.

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**You are not done until you solve each equation.**

Solve |3x – 5| = |8 + 4x|. 3x – 5 = 8 + 4x This equation sets both sides the same. This equation sets them opposite. Notice parenthesis. 3x – 5 = –(8 + 4x) or You are not done until you solve each equation.

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**The Absolute-Value Inequalities Principle**

For any positive number p and any expression x: |x| < p is equivalent to –p < x < p. (conjunction) |x| > p is equivalent to x < -p or x > p. (disjunction) Similar rules apply for less than or equal, greater than or equal.

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**So the solutions of |X| < p are those numbers that satisfy –p < X < p.**

And the solutions of |X| > p are those numbers that satisfy X < –p or p < X. ( –p p ) ( ) –p p

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**Solution Solve |x| < 3. Then graph.**

The solutions of |x| < 3 are all numbers whose distance from zero is less than 3. By substituting we see that numbers like –2, –1, –1/2, 0, 1/3, 1, and 2 are all solutions. The solution set is {x| –3 < x < 3}. In interval notation, the solution set is (–3, 3). The graph is as follows: ( )

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**In interval notation, the solution set is The graph is as follows:**

The solutions of are all numbers whose distance from zero is at least 3 units. The solution set is In interval notation, the solution set is The graph is as follows: [ ]

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**Solve |3x + 7| < 8. Then graph.**

Solution The number 3x + 7 must be less than 8 units from 0. |X| < p Replacing X with 3x + 7 and p with 8 |3x + 7| < 8 –8 < 3x + 7 < 8 –15 < 3x < 1 −5 < x < 1/3 The solution set is {x|–5 < x < 1/3}. The graph is as follows: – /3 ( )

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**Solve |3x + 7| > 8. Then graph. Solution**

The number 3x + 7 must be greater than 8 units from 0. |X| < p Replacing X with 3x + 7 and p with 8 |3x + 7| < 8 –8 < 3x + 7 < 8 –15 < 3x < 1 −5 < x < 1/3 The solution set is {x|–5 < x < 1/3}. The graph is as follows: – /3 ( )

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