a Try to rewrite this inequality as a compound inequality.
B
b Try to rewrite this inequality as a compound inequality.
C
c Create an or compound inequality because the absolute value is greater than or equal to the given value.
A
a
Solution Set:{x | -3 < x < 3} Graph:
B
b
Solution Set: {x | -2 < x < 1} Graph:
C
c
Solution Set: {x | x≤-2 or x≥ 1} Graph:
Practice makes perfect
a We are asked to find and graph the solution set for all possible values of x in the given inequality.
|x| < 3To do this, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than 3 away from the midpoint in the positive direction and any number less than 3 away from the midpoint in the negative direction.
Absolute Value Inequality:& |x| < 3
Compound Inequality:& - 3< x < 3
Graph
The graph of this inequality includes all values from -3 to 3, not inclusive. We show this by using open circles on the endpoints.
b We are asked to find and graph the solution set for all possible values of x in the given inequality.
|2x+1| < 3
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 3 away from the midpoint in the positive direction and any number less than 3 away from the midpoint in the negative direction.
Absolute Value Inequality:& |2x+1| < 3
Compound Inequality:& - 3< 2x+1 < 3We can split this compound inequality into two cases, one where 2x+1 is greater than -3 and one where 2x+1 is less than 3.
2x+1>- 3 and 2x+1 < 3
Let's isolate x in both of these cases before graphing the solution set.
This inequality tells us that all values greater than -2 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:& x < 1
Second Solution Set:& -2 < x
Intersecting Solution Set:& -2 < x < 1
Graph
The graph of this inequality includes all values from -2 to 1, not inclusive. We show this by using open circles on the endpoints.
c We are asked to find and graph the solution set for all possible values of x in the given inequality.
|2x+1|≥ 3
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 2x and -1 greater than or equal to 3 in the positive direction or in the negative direction.
2x+1 ≥ 3 or 2x+1≤ - 3Let's isolate x in both of these cases before graphing the solution set.
This inequality tells us that all values less than or equal to -2 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≥ 1
Second Solution Set:& x≤ -2
Combined Solution Set:& x≤ -2 or x≥ 1
Graph
The graph of this inequality includes all values less than or equal to -2 or greater than or equal to 1. We show this by keeping the endpoints closed.
