a Try to rewrite this inequality as a compound inequality.
B
b Create an or compound inequality, because the absolute value is greater than or equal to the given value.
C
c Try to rewrite this inequality as a compound inequality.
A
a {x | -5< x< 13 }
B
b {x | x≤-70 or x≥250}
C
c {x | 32≤ x ≤ 72}
Practice makes perfect
a We are asked to find the solution set for all possible values of x in the given absolute value inequality.
|x-4| < 9
To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than 9 away from the midpoint in the positive direction and any number less than 9 away from the midpoint in the negative direction.
Absolute Value Inequality:& |x-4| < 9
Compound Inequality:& -9 < x-4 < 9
We can split this compound inequality into two cases, one where x-4 is greater than -9 and one where x-4 is less than 9.
x-4 >-9 and x-4 < 9
Let's isolate x in both of these cases.
This inequality tells us that all values less than 13 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First Solution Set:& -5 < x
Second Solution Set:& x < 13
Intersecting Solution Set:& -5 < x < 13
b We are asked to find the solution set for all possible values of x in the given inequality.
|1/2x-45|≥ 80
To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number that makes the distance between 12x and 45 greater than or equal to 80 in the positive direction or in the negative direction.
1/2x-45 ≥ 80 or 1/2x-45≤ -80
Let's isolate x in both of these cases.
This inequality tells us that all values less than or equal to -70 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the combination of the solution sets.
First Solution Set:& x≥250
Second Solution Set:& x≤ -70
Combined Solution Set:& x≤ -70 or x≥ 250
c We are asked to find the solution set for all possible values of x in the given absolute value inequality.
|2x-5|≤ 2
To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than or equal to 2 away from the midpoint in the positive direction and any number less than or equal to 2 away from the midpoint in the negative direction.
Absolute Value Inequality:& |2x-5| ≤ 2
Compound Inequality:& -2 ≤ 2x-5 ≤ 2
We can split this compound inequality into two cases, one where 2x-5 is greater than or equal to -2 and one where 2x-5 is less than or equal to 2.
2x-5≥ -2 and 2x-5≤ 2
Let's isolate x in both of these cases.
This inequality tells us that all values less than or equal to 72 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First Solution Set:& 32 ≤ x
Second Solution Set:& x ≤ 72
Intersecting Solution Set:& 3/2 ≤ x ≤ 7/2
