Solving Inequalities With Absolute Values: A Step-by-Step Guide

Oct 21, 2025 by Dimemap Team 64 views

Hey guys! Today, we're going to dive deep into the fascinating world of inequalities, specifically those involving absolute values. If you've ever felt a bit lost when tackling these problems, don't worry – you're in the right place! We'll break down several examples step by step, so you can master this topic. Let's get started!

Understanding Absolute Value Inequalities

Before we jump into the problems, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. For example, |3| = 3 and |-3| = 3. This concept is crucial when dealing with inequalities because it means we need to consider both positive and negative scenarios. Key takeaway: Always remember that absolute value represents distance, which is always non-negative.

General Approach to Solving Absolute Value Inequalities

When you encounter an inequality like |x| < a or |x| > a, the first thing you should do is isolate the absolute value expression. This means getting the |x| part by itself on one side of the inequality. Once you've done that, you can split the problem into two separate inequalities, one for the positive case and one for the negative case.

If |x| < a, then -a < x < a.
If |x| > a, then x < -a or x > a.

Got it? Great! Now, let’s apply this to some real problems.

Problem a) 2[3|2x - 7| - 8] - 15 ≤ 23

Let's kick things off with our first inequality: 2[3|2x - 7| - 8] - 15 ≤ 23. This might look a bit intimidating, but we'll take it one step at a time.

Isolate the absolute value term. First, we need to isolate the term containing the absolute value. This means getting rid of everything outside the brackets and the absolute value signs.
- Add 15 to both sides: 2[3|2x - 7| - 8] ≤ 38
- Divide both sides by 2: 3|2x - 7| - 8 ≤ 19
- Add 8 to both sides: 3|2x - 7| ≤ 27
- Divide both sides by 3: |2x - 7| ≤ 9
Now we have |2x - 7| ≤ 9, which is much simpler to deal with. See how we systematically peeled away the layers to get to the core of the problem?
Split into two inequalities. Now that we have the absolute value isolated, we can split this into two separate inequalities:
- -9 ≤ 2x - 7 ≤ 9
This compound inequality is key. It tells us that 2x - 7 must be between -9 and 9, inclusive.
Solve each inequality. Let's solve this compound inequality:
- Add 7 to all parts: -2 ≤ 2x ≤ 16
- Divide all parts by 2: -1 ≤ x ≤ 8
So, the solution to the first inequality is -1 ≤ x ≤ 8. This means any value of x between -1 and 8 (including -1 and 8) will satisfy the original inequality.

Problem b) 3[3|2x + 13| - 12] - 19 < 26

Next up, we have 3[3|2x + 13| - 12] - 19 < 26. Let's follow the same steps as before.

Isolate the absolute value term:
- Add 19 to both sides: 3[3|2x + 13| - 12] < 45
- Divide both sides by 3: 3|2x + 13| - 12 < 15
- Add 12 to both sides: 3|2x + 13| < 27
- Divide both sides by 3: |2x + 13| < 9
We've successfully isolated the absolute value: |2x + 13| < 9.
Split into two inequalities:
- -9 < 2x + 13 < 9
Solve each inequality:
- Subtract 13 from all parts: -22 < 2x < -4
- Divide all parts by 2: -11 < x < -2
Therefore, the solution to this inequality is -11 < x < -2.

Problem c) 2[3|2x + 11| - 9] - 7 ≤ 5

Now, let's tackle 2[3|2x + 11| - 9] - 7 ≤ 5.

Isolate the absolute value term:
- Add 7 to both sides: 2[3|2x + 11| - 9] ≤ 12
- Divide both sides by 2: 3|2x + 11| - 9 ≤ 6
- Add 9 to both sides: 3|2x + 11| ≤ 15
- Divide both sides by 3: |2x + 11| ≤ 5
Split into two inequalities:
- -5 ≤ 2x + 11 ≤ 5
Solve each inequality:
- Subtract 11 from all parts: -16 ≤ 2x ≤ -6
- Divide all parts by 2: -8 ≤ x ≤ -3
The solution is -8 ≤ x ≤ -3.

Problem d) 3[2|2x + 3| - 9] - 8 < 7

Let's move on to 3[2|2x + 3| - 9] - 8 < 7.

Isolate the absolute value term:
- Add 8 to both sides: 3[2|2x + 3| - 9] < 15
- Divide both sides by 3: 2|2x + 3| - 9 < 5
- Add 9 to both sides: 2|2x + 3| < 14
- Divide both sides by 2: |2x + 3| < 7
Split into two inequalities:
- -7 < 2x + 3 < 7
Solve each inequality:
- Subtract 3 from all parts: -10 < 2x < 4
- Divide all parts by 2: -5 < x < 2
The solution is -5 < x < 2.

Problem e) 2[3|2x - 3| - 8] - 9 < 5

Let's keep going with 2[3|2x - 3| - 8] - 9 < 5.

Isolate the absolute value term:
- Add 9 to both sides: 2[3|2x - 3| - 8] < 14
- Divide both sides by 2: 3|2x - 3| - 8 < 7
- Add 8 to both sides: 3|2x - 3| < 15
- Divide both sides by 3: |2x - 3| < 5
Split into two inequalities:
- -5 < 2x - 3 < 5
Solve each inequality:
- Add 3 to all parts: -2 < 2x < 8
- Divide all parts by 2: -1 < x < 4
The solution is -1 < x < 4.

Problem f) 2[3|2x - 5| - 4] - 7 ≤ 3

Now, let's tackle 2[3|2x - 5| - 4] - 7 ≤ 3.

Isolate the absolute value term:
- Add 7 to both sides: 2[3|2x - 5| - 4] ≤ 10
- Divide both sides by 2: 3|2x - 5| - 4 ≤ 5
- Add 4 to both sides: 3|2x - 5| ≤ 9
- Divide both sides by 3: |2x - 5| ≤ 3
Split into two inequalities:
- -3 ≤ 2x - 5 ≤ 3
Solve each inequality:
- Add 5 to all parts: 2 ≤ 2x ≤ 8
- Divide all parts by 2: 1 ≤ x ≤ 4
The solution is 1 ≤ x ≤ 4.

Problem g) 2[3|2x + 1| - 5] - 9 < 10

Last but not least, let's solve 2[3|2x + 1| - 5] - 9 < 10.

Isolate the absolute value term:
- Add 9 to both sides: 2[3|2x + 1| - 5] < 19
- Divide both sides by 2: 3|2x + 1| - 5 < 9.5
- Add 5 to both sides: 3|2x + 1| < 14.5
- Divide both sides by 3: |2x + 1| < 14.5 / 3
- Simplify the fraction: |2x + 1| < 4.8333...
Split into two inequalities:
- -4.8333... < 2x + 1 < 4.8333...
Solve each inequality:
- Subtract 1 from all parts: -5.8333... < 2x < 3.8333...
- Divide all parts by 2: -2.9166... < x < 1.9166...
So, the solution to this inequality is approximately -2.92 < x < 1.92.

Conclusion

And there you have it! We've tackled seven different absolute value inequalities, and hopefully, you're feeling much more confident about solving these types of problems. Remember the key steps: isolate the absolute value, split into two inequalities, and solve each one separately. With a bit of practice, you'll be solving these like a pro. Keep up the great work, guys!