Download Writing absolute value equations and inequalities

Writing absolute value equations and inequalities
Name: ______________________________
You know how to solve an absolute value equation, and you know how to solve an absolute value
inequality. You even know how to change an absolute value function into a piecewise function. But
how do you write the absolute value function to start with?
Think of what an absolute value equation really means. In words an absolute value equation might
be:
What two numbers are 4 away from 16?
You know immediately that the answers are 12 and 20. But what does that equation look like?
Use the number that you are starting on as if it were the vertex of your absolute value. In this case
the vertex would be (16, 0). To put this into an absolute value equation you would have x  16 +0.
The +0 is unnecessary so you have x  16 . **Remember that the absolute value equation always
has the opposite sign of the x-value of the vertex (positive or negative).
Then simply set it equal to how far away you are. x  16 =4
Example 2: What equation represents all points exactly 2 units from 3?
The vertex is (3,0) so the equation would be x  3 =2.
Example 3: What equation represents all points exactly 12 units from -7?
The vertex is (-7, 0) so the equation would be x  7 =12.
Try a few:
1. What equation represents all points at are 15 units from 3?
2. What equation represents all points at are 3 units from -6?
3. What equation represents all points at are 4 units from 52?
4. What equation represents all points at are 7 units from -30?
Writing inequalities is done the same way, but instead of an equal sign you have an inequality.
For example, A bed must be within 1 feet of 9 feet. Write an absolute value inequality below that
represents all possible lengths of a box.
If it ever says within, vary, or between two numbers, your inequality sign will always say the
absolute value is less than the number.
This equation would be x  9  1.
Example 4: The average price for a music CD is $15.50. Depending on where you shop, the price
may vary by as much as $3.00. Write an absolute value inequality describing the possible prices of
music CD’s.
The equation would be x 15.50  3.00
Example 5: Your car averages 35 miles per gallon on the highway. The actual mileage varies from
the average by 5 miles per gallon. Write an absolute value inequality that shows the range for the
mileage your car gets.
The equation would be x  35  5
Try a few:
1. The average price of a particular brand of shampoo is $3.26. Depending on where you shop, the
price may vary by as much as $0.25. Write an absolute value inequality describing the possible
prices of the shampoo. Solve the inequality.
2. Physicians consider an adult’s body temperature to be normal if it is 98.6  F, plus or minus 1  F.
Write an absolute value inequality that describes this normal temperature range.
3. In woodshop class, you must cut a piece of wood within
3
inch of the teacher’s specification of
16
2
inches. Write an absolute value inequality that describes the acceptable lengths for this piece of
16
wood.
5