absolute value equalities and inequalities candace moraczewski and greg fisher © april, 2004
TRANSCRIPT
ABSOLUTE VALUE EQUALITIES and INEQUALITIES
Candace Moraczewski and Greg Fisher
© April, 2004
3 x
-3 3
This absolute value equation represents the numbers on the number line whose distance from 0 is equal to 3.
0
3 units 3 units
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.
An absolute value equation is an equation that containsa variable inside the absolute value sign.
The absolute value of a number is its distance from 0 on a number line.
-5 0
5 5- because -5 is 5 units from 0
-3
3 3- because -3 is 3 units from 0
Absolute Value Equalities
Solve | x | = 7
x = 7 or x=-7
{-7, 7}
Solve | x +2| = 7
x +2= 7 or x+2=-7
{5,-9}
x=5 or x = -9
Solve 4|x – 3| + 2 = 104| x – 3 | = 8
| x – 3 | = 2
x – 3 = 2 or x-3 = -2
x = 5 or x= 1{1,5}
Solve -2|2x + 1|-3 = 9
-2| 2x + 1| = 12
| 2x + 1| = -6
NO SOLUTION Because Abs. value cannot be negative
0
Pause!
• Try 1-4 on Absolute Value Worksheet
MEMORIZE THIS:
• GreatOR• Or statement, two inequalities
• Less THAND• Sandwich, one inequality two signs
-3 3
0
x
If a number x is between -3 and 3 then this translates to:
Inequality notation: -3 < x < 3 (a double inequality)
Absolute value notation: 3 x
because -3 is to the left of x and x is to the left of 3
because all of the numbers between -3 and 3 have adistance from 0 less than 3
-3 3
0
x
If a number x is between -3 and 3, including the -3 and 3,then this translates to:
Inequality notation: -3 x 3 (a double inequality)
Absolute value notation: 3 x
-3 3
0
x
If a number x is to the left of -3 or to the right of 3 thenthis translates to:
Inequality notation: x < -3 or x > 3 (a compound “or” inequality)
Absolute value notation: 3 x
x
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 adistance from 0 greater than 3
because x is to the left of -3 or x is to the right of 3
-
-3 3
0
x
If a number x is to the left of -3 or to the right of 3, includingthe -3 and 3, then this translates to:
Inequality notation: x -3 or x 3 (a compound “or” inequality)
Absolute value notation: 3 x
x
-
2 x 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.
0 -2 2
Inequality notation: -2 < x < 2
x
2 x
0 -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: 2x2
x
2 x
0 -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.
Inequality notation: x < -2 or x > 2
x x
-
2 x
0 -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:
2or x -2x
x x
-
TRY THE FOLLOWING PROBLEMS, CHECK YOUR ANSWERS WITH A PARTNER
2 3x - 4 .6
1 1 2x .5
1 2x - 7- .4
5 3x - 2- .3
4 3 - x .2
5 3 - 2 .1
x
Solve the following absolute value inequalities. Write answer using both inequality notation and interval notation.
ANSWERS:
] 4 1,- [ , 4 x 1- .1
Click here to returnto the problem set
) [7, 1]- ,- ( , 7 or x 1- x 2.
ANSWERS:
Click here to returnto the problem set
) 1 , 37- ( , 1 x
37- 3.
ANSWERS: Click here to returnto the problem set
) 3,- [ 4]- ,- ( , 3- or x 4- x 4.
ANSWERS:
Click here to returnto the problem set
0) 1,- ( , 0 x 1- 5.
ANSWERS:Click here to returnto the problem set
) 2, [ ] 32 ,- ( , 2 or x
32 x 6.
ANSWERS:
Click here to returnto the problem set
Pause!
• Try 5-8 on Absolute Value Worksheet on your own
Can the absolute value of something be less than zero?
• NO! Absolute value is always positive.
• Cases:
512 xAll real numbers. The
absolute value will always be greater than zero.
38 x No solution. The absolute value will never be less than zero. Just like absolute value
cannot be = to a negative number.
Pause!
• More practice is on the back
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.
NOTATION: (old) 2 ≤ x < 3 (new) x ≥ 2 x < 3 0 1 2 3
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 0 1 3 4 52NOTATION: (old) x ≤ -1 or x > 4 (new) x ≤ -1 x > 4
Recap
• Intersection: AND, , overlap • Union: OR, , opposite directions
• Always write answers small to big (left to right)
“U” for Union
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 +4 +4 +4
9 < x < 6
Add four to each “side”
Ex: -16 < 5 – 3q < 11
- 5 -5 -5
-21 < -3q < 6
**Remember flip the sign if you multiply or divide by a negative number!-3 -3 -3
7 > q > -2 Rewrite…. -2 < q < 7
Pause!
• Answer 5-8 on page 6 in workbook (section 1.6)
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
Solve 4 1 -x 2
0-4 4
2x - 1
4 1-2x 4
Now solve the double inequality
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
4 1 -2x 4
Divide every position by 2
25 x
23
+1 +1 +1 ________________
5 2x 3
Solve 8 2 3x
0-8 8
3x + 2
8 2 3x 8
Now solve the double inequality
3. Translate this to either a double inequality or two inequalities going in opposite directions
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)
8 2 3x 8
Divide every position by 3
2 x 310
-2 -2 -2 ________________
6 3x 10
Solve 5 2 x
0-5 5
x + 2
5 2 or x 5- 2 x
Now solve the “or” compound inequality
x + 2
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
-
5 2 or x 5- 2 x -2 -2 -2 -2
3 or x 7- x
Solve 2 3x - 4
0-2 2
4 – 3x
2 3x 4or 2- 3x - 4
Now solve the “or” compound inequality
4 – 3x
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
-
-4 -4 -4 -4
2- 3x -or 6- 3x -
2 3x 4or 2- 3x - 4
Divide both inequalities by -3. Remember to changethe sense of the inequality signs because of divisionby a negative.
32 or x 2 x
Pause!
• Answer 9-16 in your workbook (pg 6)
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> -9x<74 AND x >56 56<x<74
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 > -.4x < 6.4 AND x > 5.6 5.6< x < 6.4
• 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 RangeCheck answer: x-78 < 3 AND x-78>-3
x<81 AND x>75 75<x<81
Pause!
• Try word problems from overhead