What are all values of x for which |2x-3| < 7 ?
1, 2, 3, 4
If 2x-3 > 0, which means x >or 1.5,
Ms Sue has provided some but not all possible values of x.
First assume 2x-3 >0
This requires x > 1.5, and
2x-3 < 7
That means x < 5.
Therefore 1.5 < x < 5 are answers.
If 2x < 3, which means x < 1.5,
3 - 2x < 7
x > -2
Therefore -2 < x < 1.5 are answers
Put them together and you find that -2 < x < 5 is the range of answers.
To find all values of x for which the absolute value of 2x-3 is less than 7, we can use the definition of the absolute value.
For any real number a, |a| is defined as:
- a if a ≥ 0
- -a if a < 0
We can apply the same definition to the expression |2x-3| < 7.
If 2x-3 > 0, then |2x-3| = 2x-3
If 2x-3 < 0, then |2x-3| = -(2x-3) = -2x+3
Now, we can solve the inequality |2x-3| < 7 by considering the two cases:
Case 1: 2x-3 > 0
If 2x - 3 > 0, then the inequality becomes:
2x - 3 < 7
Solving this inequality for x, we have:
2x < 10
x < 5
Now, we need to check if this solution satisfies the original inequality.
Substituting x < 5 into the original inequality:
|2x-3| < 7
|2(5)-3| < 7
|10-3| < 7
|7| < 7
7 < 7
Since 7 is not less than 7, the solution x < 5 does not satisfy the original inequality.
Case 2: 2x-3 < 0
If 2x - 3 < 0, then the inequality becomes:
-(2x - 3) < 7
-2x + 3 < 7
Solving this inequality for x, we have:
-2x < 4
x > -2
Now, we need to check if this solution satisfies the original inequality.
Substituting x > -2 into the original inequality:
|2x-3| < 7
|2(-2)-3| < 7
|-4-3| < 7
|-7| < 7
7 < 7
Since 7 is not less than 7, the solution x > -2 does not satisfy the original inequality.
Therefore, there are no values of x that satisfy the inequality |2x-3| < 7.