Section 4.3 Absolute Value Equations & Inequalities  Absolute Value Equations |2x + 5| = 13  Absolute Value Inequalities |3x – 2| <

Find the Solution Set for |x| =

Find the Solution Set for |x| = 0 Find the Solution Set for |x| =

Using the Absolute Value Principle  |x + 1| = 2 x + 1 = 2 or x + 1 = -2 {1,-3}  |2y – 6| = 0 2y – 6 = 0 {3}  |5x – 3| = -2 no solution φ 4.34

Solve 4.35

Solve 4.36

Solve 4.37

If f(x) = 2|x + 3| + 1 find all x for which f(x) = 15 First, You must Isolate the Radical 4.38

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What do we do when an equation has Two Absolute Values?  |3| = |3| or |3| = |-3| or |-3| = |3| or |-3| = |-3|  Only 2 cases: They are equal, or opposite  If |X| = |Y| then solve X = Y or X = -Y  Let’s try it: |5x + 3| = |3x + 25|  5x + 3 = 3x + 25 or 5x + 3 = -(3x + 25)  or 5x + 3 = -3x – 25  2x = 22 or 8x = -28  x = 11 or x = -7/

Solve 4.311

Opposites Practice  |x + 4| = |2x – 7| x + 4 = 2x – 7 or x + 4 = -2x + 7  |x + 4| = |x – 3| x + 4 = x – 3 or x + 4 = -x + 3  |3a – 1| = |2a + 4| 3a – 1 = 2a + 4 or 3a – 1 = -2a – 4  |n – 3| = |3 – n| n – 3 = 3 – n or n – 3 = -3 + n  |7 – a| = |a + 4| 7 – a = a + 4 or 7 – a = -a –

Inequalities with Absolute Value ) ( 4.313

A Similar Example [ ] ][ 4.314

Principles for Solving AVE’s ( ) )( 4.315

Solve -⅔ )( -p < X < p 4.316

If f(x)=|4x + 2| then find all x such that f(x) ≥ 6 [ ] X ≤ -p or p ≤ X 4.317

What if p is a Negative Number?  Consider |x + 3| < -5 (or ≤ )  That’s impossible … so: No Solution ϕ  Consider |x + 3| > -5 (or ≥ )  The absolute value will always be greater, so: All Real Numbers R 4.318

What Next?  Section Wednesday Intro to Polynomials & Polynomial Functions Section