Absolute Value Equations & Inequalities

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Solving Absolute Value Inequalities

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Presentation transcript:

Absolute Value Equations & Inequalities EQ: How does solving an absolute value equation/inequality compare to solving a linear equation/inequality?

I am 3 “miles” from my house |B| = 3 B = 3 or B = -3

Review Ex 1) |7| = Ex 2) |-5| = Ex 3) 5|2 – 4| + 2 = Absolute Value: the distance a number is from zero (always positive) Ex 1) |7| = Ex 2) |-5| = Ex 3) 5|2 – 4| + 2 =

Absolute Value Equations Ex 1) Solve: |x| = 8 Ex 2) Solve: |x| = 25 Ex 3) Solve: |x| = -10

Have two solutions because the absolute value of a positive number is the same as the absolute value of a negative number.

Solving Absolute Value Equations Step 1: Get “bars” alone on one side Step 2: Re-write as two equations; flip the signs on the RIGHT side (drop the bars) Step 3: Solve both equations Step 4: Check for an extraneous solution!!

Extraneous Solution A value you get after correctly solving the problem that does not actually satisfy the equation.

Ex 4) 3|x + 2| - 7 = 14 3|x + 2| = 21 |x + 2| = 7

|x + 2| = 7 x + 2 = 7 x + 2 = -7 x = 5 or x = -9

Check Your Answers: x = 5 x = -9 3|x + 2| - 7 = 14 3|x + 2| - 7 = 14 3|5 + 2| - 7 = 14 3|-9 + 2| - 7 = 14 3|7| - 7 = 14 3|-7| - 7 = 14 14 = 14 14 = 14

Ex 5) |3x + 2| = 4x + 5 3x + 2 = 4x + 5 3x + 2 = -4x – 5

3x + 2 = -4x – 5 3x + 2 = 4x + 5

Check Your Solutions |3x + 2| = 4x + 5

|x – 4| + 7 = 2 |x – 4| = -5 NO SOLUTION Ex 6) |x – 4| + 7 = 2 |x – 4| = -5 NO SOLUTION

Solving by Graphing Step 1: Enter the left & right side into Y1 & Y2 PRESS → NUM #1 abs( Step 2: Find the first intersection Step 3: Find the second intersection if there is one MATH #5 2nd TRACE

3|x + 2| -7 = 14

|3x + 2| = 4x + 5

Solve: |5x| + 10 = 55 Solve: |x – 3| = 10 Solve: 2|y + 6| = 8 Practice Solve: |5x| + 10 = 55 Solve: |x – 3| = 10 Solve: 2|y + 6| = 8 Solve: |a – 5| + 3 = 2 Solve: |4x + 9| = 5x + 18

Thank about it… If |x| < 5 what are some possible values of x? If |x| > 5 what are some possible values of x?

Solving Absolute Value Inequalities Since the inequalities are absolute value, there are still going to be two solutions When writing the second equation, be sure to flip the inequality sign. Also, when dividing by a negative the inequality sign flips.

Ex 1)

Ex 2)

You Try! Solve | 2x + 3 | < 6

Ex 3) Solve

You Try! Solve |2x-5|>x+1.

Quick Write: Think of some objects or situations that need to be within a certain “value” – if you go too much over it would be a bad thing, and if you go too much under it would also be a bad thing:

Tolerance There are strict height requirements to be a “Rockette” You must be between 66 inches 70.5 inches

Tolerance: The difference between a desired measurement and its maximum and minimum allowable values. Most Allowed Least Allowed Perfect Amount Tolerance Tolerance

Tolerance: __________ Ex 1) The doctor says that you need to stay between 125 and 135 lbs to be at a healthy weight. Min: _______ Perfect: _______ Max: ________ Tolerance: __________

Example 2) Workers at a hardware store take their morning break no earlier than 10 am and no later than noon. Let c represent the time the workers take their break. Write an absolute value inequality to represent the situation.

Exit Ticket: How is solving an absolute value inequality different than solving an absolute value equation? What is the difference between the solutions of an equation and the solutions of an inequality?