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ABSOLUTE VALUE EQUALITIES and INEQUALITIES
Candace Moraczewski and Greg Fisher © April, 2004
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An absolute value equation is an equation that contains
a variable inside the absolute value sign. This absolute value equation represents the numbers on the number line whose distance from 0 is equal to 3. Two numbers satisfy this equation. Both 3 and -3 are 3 units from 0. Look at the number line and notice the distance from 0 of -3 and 3. 3 units 3 units -3 3
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The absolute value of a number is its distance from 0 on a number line.
-5 -3 because -5 is 5 units from 0 because -3 is 3 units from 0
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Absolute Value Equalities
Solve | x | = 7 x = 7 or x=-7 {-7, 7}
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Solve | x +2| = 7 x +2= 7 or x+2=-7 x=5 or x = -9 {5,-9}
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4| x – 3 | = 8 | x – 3 | = 2 x – 3 = 2 or x-3 = -2 x = 5 or x= 1 {1,5}
Solve 4|x – 3| + 2 = 10 4| x – 3 | = 8 | x – 3 | = 2 x – 3 = 2 or x-3 = -2 x = 5 or x= 1 {1,5}
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Abs. value cannot be negative
Solve -2|2x + 1|-3 = 9 -2| 2x + 1| = 12 | 2x + 1| = -6 NO SOLUTION Because Abs. value cannot be negative
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Try 1-4 on Absolute Value Worksheet
Pause! Try 1-4 on Absolute Value Worksheet
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MEMORIZE THIS: GreatOR Less THAND Or statement, two inequalities
Sandwich, one inequality two signs
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x -3 3 If a number x is between -3 and 3 then this translates to: Absolute value notation: because all of the numbers between -3 and 3 have a distance from 0 less than 3 Inequality notation: < x < 3 (a double inequality) because -3 is to the left of x and x is to the left of 3
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x -3 3 If a number x is between -3 and 3, including the -3 and 3, then this translates to: Absolute value notation: Inequality notation: x (a double inequality)
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because x is to the left of -3 or x is to the right of 3
3 If a number x is to the left of -3 or to the right of 3 then this translates to: Absolute value notation: because the numbers to the left of -3 have a distance from 0 greater than 3 and the numbers to the right of 3 have a distance from 0 greater than 3 Inequality notation: x < -3 or x > 3 (a compound “or” inequality) because x is to the left of -3 or x is to the right of 3
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x x -3 3 If a number x is to the left of -3 or to the right of 3, including the -3 and 3, then this translates to: Inequality notation: x or x (a compound “or” inequality) Absolute value notation:
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This absolute value inequality represents all of the numbers on a number line whose distance from 0 is less than 2. See the red shaded line below. x -2 2 -2 < x < 2 Inequality notation:
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x -2 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is less than or equal to 2. Notice that both -2 and 2 are included on this interval. Inequality notation:
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x x -2 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is more than 2. Notice that the intervals satisfying this inequality are going in opposite directions. x < -2 or x > 2 Inequality notation:
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x x -2 2 This absolute value inequality represents all of the numbers on the number line whose distance from 0 is more than or equal to 2. Notice that the intervals satisfying this inequality are going in opposite directions and that 2 and -2 are included on the intervals. Inequality notation:
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TRY THE FOLLOWING PROBLEMS, CHECK YOUR ANSWERS WITH A PARTNER
Solve the following absolute value inequalities. Write answer using both inequality notation and interval notation.
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ANSWERS: Click here to return to the problem set
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ANSWERS: Click here to return to the problem set
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ANSWERS: Click here to return to the problem set
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ANSWERS: Click here to return to the problem set
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ANSWERS: Click here to return to the problem set
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ANSWERS: Click here to return to the problem set
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Try 5-8 on Absolute Value Worksheet on your own
Pause! Try 5-8 on Absolute Value Worksheet on your own
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Can the absolute value of something be less than zero?
NO! Absolute value is always positive. Cases: All real numbers. The absolute value will always be greater than zero. No solution. The absolute value will never be less than zero. Just like absolute value cannot be = to a negative number.
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More practice is on the back
Pause! More practice is on the back
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Compound Inequalities
Contains 2 parts 1. Intersection: intersection is a compound inequality that contains AND. The solution must be a solution of BOTH inequalities to be true in the compound inequality Ex: Graph the solution set of x < 3 and x ≥ 2. 1 2 3 NOTATION: (old) 2 ≤ x < 3 (new) x ≥ x < 3
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Compound Inequalities cont’d
2. Union: intersection is a compound inequality that contains OR. The solution must be a solution of EITHER inequality to be true in the compound inequality Ex: Graph the solution set of x ≤ -1 or x > 4. -2 -1 1 2 3 4 5 NOTATION: (old) x ≤ -1 or x > 4 (new) x ≤ x > 4
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Recap “U” for Union Intersection: AND, , overlap
Union: OR, , opposite directions Always write answers small to big (left to right) “U” for Union
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How to solve compound inequalities
Think of it as solving two different inequalities and then combine their solutions as an intersection. Ex: -5 < x – 4 < 2 9 < x < 6 Add four to each “side” **Remember flip the sign if you multiply or divide by a negative number! Ex: -16 < 5 – 3q < 11 -21 < q < -3 -3 -3 7 > q > Rewrite… < q < 7
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Pause! Answer 5-8 on page 6 in workbook (section 1.6)
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TO SOLVE A MORE COMPLICATED ABSOLUTE VALUE INEQUALITY, FOLLOW THESE STEPS AS ILLUSTRATED IN THE FOLLOWING EXAMPLES 1. Draw a number line and identify the interval(s) which satisfy the inequality 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions connected with the word “or” 4. Remember to include the endpoint if the inequality also has an equal to symbol
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Now solve the double inequality
1. Draw a number line and identify the interval(s) which satisfy the inequality: 2x - 1 -4 4 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the double inequality
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Divide every position by 2
________________ Divide every position by 2
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Now solve the double inequality
1. Draw a number line and identify the interval(s) which satisfy the inequality 3x + 2 8 -8 2..Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the double inequality
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Divide every position by 3
________________ Divide every position by 3
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Now solve the “or” compound inequality
1. Draw a number line and identify the interval(s) which satisfy the inequality x + 2 x + 2 -5 5 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the “or” compound inequality
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Now solve the “or” compound inequality
1. Draw a number line and identify the interval(s) which satisfy the inequality 4 – 3x 4 – 3x -2 2 2. Write the expression in the absolute value sign over the designated interval(s) 3. Translate this to either a double inequality or two inequalities going in opposite directions Now solve the “or” compound inequality
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Divide both inequalities by -3. Remember to change the sense of the inequality signs because of division by a negative.
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Pause! Answer 9-16 in your workbook (pg 6)
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|x – 65| < 9 Word Problems
Pretend that you are allowed to go within 9 of the speed limit of 65mph without getting a ticket. Write an absolute value inequality that models this situation. |x – 65| < 9 Desired amount Acceptable Range Check Answer: x-65< 9 AND x-65> -9 x<74 AND x >56 56<x<74
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|x – 6| < .4 Word Problems
If a bag of chips is within .4 oz of 6 oz then it is allowed to go on the market. Write an inequality that models this situation. |x – 6| < .4 Desired amount Acceptable Range Check Answer: x – 6 < .4 AND x – 6 > -.4 x < AND x > < x < 6.4
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In a poll of 100 people, Misty’s approval rating as a dog is 78% with a 3% of error. ticket. Write an absolute value inequality that models this situation. |x – 78| < 3 Desired amount Acceptable Range Check answer: x-78 < 3 AND x-78>-3 x<81 AND x>75 75<x<81
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Pause! Try word problems from overhead
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