We are given the inequality:
>
|3x - 5| - |2x - 7| ≥ 6
This is an
absolute value inequality with two absolute value expressions. To solve it, we need to consider the
critical points where the expressions inside the absolute values change sign — these are the points where each expression equals zero.
---
Step 1: Find Critical Points
Set each expression inside the absolute values equal to zero:
-
3x - 5 = 0 → x = 5/3 ≈ 1.666...
-
2x - 7 = 0 → x = 7/2 = 3.5
These two critical points divide the real number line into
three intervals:
1.
x < 5/3
2.
5/3 ≤ x < 3.5
3.
x ≥ 3.5
We will solve the inequality in each interval by removing the absolute values appropriately (based on the sign of the expression in that interval).
---
## 🔹 Interval 1: x < 5/3
In this region:
- 3x - 5 < 0 → |3x - 5| = -(3x - 5) = -3x + 5
- 2x - 7 < 0 → |2x - 7| = -(2x - 7) = -2x + 7
Substitute into the inequality:
> (-3x + 5) - (-2x + 7) ≥ 6
> -3x + 5 + 2x - 7 ≥ 6
> -x - 2 ≥ 6
> -x ≥ 8
>
x ≤ -8
Now, check if this solution lies within the current interval (
x < 5/3). Since
x ≤ -8 is certainly less than 5/3, this is valid.
✔ Solution from Interval 1: x ≤ -8
---
## 🔹 Interval 2: 5/3 ≤ x < 3.5
In this region:
- 3x - 5 ≥ 0 → |3x - 5| = 3x - 5
- 2x - 7 < 0 → |2x - 7| = -2x + 7
Substitute:
> (3x - 5) - (-2x + 7) ≥ 6
> 3x - 5 + 2x - 7 ≥ 6
> 5x - 12 ≥ 6
> 5x ≥ 18
>
x ≥ 18/5 = 3.6
But wait — our current interval is
x < 3.5, and 3.6 > 3.5. So
x ≥ 3.6 does
not lie in this interval.
✘ No solution in Interval 2
---
## 🔹 Interval 3: x ≥ 3.5
In this region:
- 3x - 5 > 0 → |3x - 5| = 3x - 5
- 2x - 7 ≥ 0 → |2x - 7| = 2x - 7
Substitute:
> (3x - 5) - (2x - 7) ≥ 6
> 3x - 5 - 2x + 7 ≥ 6
> x + 2 ≥ 6
>
x ≥ 4
Check if this lies in the current interval (
x ≥ 3.5). Yes,
x ≥ 4 is within this interval.
✔ Solution from Interval 3: x ≥ 4
---
##
✔ Final Answer:
Combine the solutions from all intervals:
>
x ≤ -8 or
x ≥ 4
In interval notation:
>
(-∞, -8] ∪ [4, ∞)
---
## 📌 Verification (Optional but Recommended)
Let’s test a point in each solution region.
-
Test x = -9 (in x ≤ -8):
|3(-9) - 5| - |2(-9) - 7| = |-27 -5| - |-18 -7| = | -32 | - | -25 | = 32 - 25 = 7 ≥ 6
✔
-
Test x = 5 (in x ≥ 4):
|15 - 5| - |10 - 7| = |10| - |3| = 10 - 3 = 7 ≥ 6
✔
-
Test x = 3 (in gap between -8 and 4):
|9 - 5| - |6 - 7| = 4 - 1 = 3 < 6
✘ — not part of solution, as expected.
---
##
✔ Final Answer:
>
x ≤ -8 or x ≥ 4
Or in interval notation:
>
(-∞, -8] ∪ [4, ∞)
Parent Tip: Review the logic above to help your child master the concept of solve absolute value inequalities worksheet.
