|2x+1| < 5
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 5 away from the midpoint in the positive direction and any number less than 5 away from the midpoint in the negative direction.
Type of Inequality
Inequality
Absolute Value
|2x+1| < 5
Compound
- 5< 2x+1 < 5
We can split this compound inequality into two cases, one where 2x+1 is greater than -5 and one where 2x+1 is less than 5.
2x+1>- 5 and 2x+1 < 5
Let's isolate x in both of these cases.
This inequality tells us that all values greater than -3 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 < 2
Second Solution Set:& -3 < x
Intersecting Solution Set:& -3 < x < 2
b Similarly as to Part A, we will rewrite the given inequality as a compound inequality. Before we do that, let's divide both sides of the inequality by 2 to isolate the absolute value expression.
2|3x-2|≥ 10 ⇔ |3x-2|≥ 5
Now, we can create a compound inequality by removing the absolute value. In this case, the solution set contains any number that makes the distance between 3x and 2 greater than or equal to 5 in the positive direction or in the negative direction.
3x-2 ≥ 5 or 3x-2≤ - 5
Let's isolate x in both of these cases.
This inequality tells us that all values less than or equal to - 1 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≥ 7/3 [0.5em]
Second Solution Set:& x≤ -1 [0.5em]
Combined Solution Set:& x ≤ -1 or x ≥ 73
