Solving Absolute Value Inequalities Activity Maze - Free Printable

Let’s solve this maze step by step.

We start at the box labeled “START HERE!” which has the inequality:
|x| > 8

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Step 1: Solve |x| > 8

Absolute value inequalities like |x| > a mean that x is more than a units away from zero — so it’s either greater than a or less than -a.

So, |x| > 8 → x > 8 or x < -8

Now look at the paths coming out of the START box. We need to find the path that says “x > 8 or x < -8”.

Looking at the maze, right next to the START box, there’s a small rectangle with:

> x > 8 or x < -8

That’s our answer! So we follow that path.

It leads us to the box:
|x| ≤ 2

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Step 2: Solve |x| ≤ 2

This means x is within 2 units of zero — including -2 and 2.

So, |x| ≤ 2 → -2 ≤ x ≤ 2

Look for the path labeled “-2 ≤ x ≤ 2” coming out of this box.

Yes — directly to the right, there’s a path that says:

> -2 ≤ x ≤ 2

Follow that path to the next box:
|8x| < 24

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Step 3: Solve |8x| < 24

First, divide both sides by 8:

|x| < 3

Which means:
-3 < x < 3

Now check the paths coming out of |8x| < 24.

One path says: x < 3 — too vague.

Another says: -3 < x < 3 ← That’s exactly what we want!

Follow that diagonal path down-left to the next box:
-7|x - 8| < -56

Wait — let me double-check the direction. Actually, looking again:

From |8x| < 24, the correct path should be the one matching our solution: -3 < x < 3

In the maze, from |8x| < 24, going diagonally down-left, there’s a label:

> -3 < x < 3

Perfect. That leads to the box:
-7|x - 8| < -56

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Step 4: Solve -7|x - 8| < -56

First, divide both sides by -7. BUT — remember: when you divide or multiply an inequality by a negative number, you must flip the inequality sign.

So:

-7|x - 8| < -56
→ Divide by -7 → flip < to >
→ |x - 8| > 8

Now solve |x - 8| > 8

This means:
x - 8 > 8 OR x - 8 < -8

Solve each:

x - 8 > 8 → x > 16
x - 8 < -8 → x < 0

So overall: x > 16 or x < 0

Now look at the paths coming out of this box.

There’s a path labeled:
> x > 16 or x < 0

Exactly matches! Follow that path (diagonally up-right) to the next box:
4|x + 2| ≥ 16

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Step 5: Solve 4|x + 2| ≥ 16

Divide both sides by 4:

|x + 2| ≥ 4

This means:
x + 2 ≥ 4 OR x + 2 ≤ -4

Solve:

x + 2 ≥ 4 → x ≥ 2
x + 2 ≤ -4 → x ≤ -6

So: x ≥ 2 or x ≤ -6

Check paths from this box.

One path says:
> x ≥ 2 or x ≤ -6? Wait — actually, in the maze, from 4|x+2|≥16, the path going left-down says:

> x ≥ 2 or x ≤ -6? Let me recheck.

Actually, looking carefully — from 4|x+2|≥16, the path going down-left is labeled:

> x ≥ 2 or x ≤ -6? Hmm, maybe not written exactly.

Wait — perhaps I made a mistake earlier. Let me trace back.

Actually, after solving |x - 8| > 8 → x > 16 or x < 0, and following that path, we arrive at 4|x + 2| ≥ 16.

But now, solving 4|x + 2| ≥ 16 gives |x + 2| ≥ 4 → x ≥ 2 or x ≤ -6.

Is there a path labeled that? Looking at the maze...

Actually, from 4|x+2|≥16, the path going down-left goes to |x + 6| - 7 < 4? No.

Wait — perhaps I misread the connection.

Let me try a different approach. Maybe I should list all boxes and their connections logically.

Alternatively, let’s continue assuming we’re on track.

After 4|x+2|≥16, if we go to the path that matches our solution, but since we don’t see exact match, perhaps we made an error.

Wait — let’s recalculate Step 4.

Step 4: -7|x - 8| < -56

Divide by -7 → flip inequality → |x - 8| > 8 → correct.

Then x - 8 > 8 → x > 16; x - 8 < -8 → x < 0 → correct.

Now, from that box, the path labeled “x > 16 or x < 0” leads to 4|x + 2| ≥ 16 — yes.

Now, solving 4|x + 2| ≥ 16 → |x + 2| ≥ 4 → x ≥ 2 or x ≤ -6.

Now, look at the paths from 4|x + 2| ≥ 16.

In the maze, from that box, there are two paths:

One going down-left: labeled “x ≥ 2 or x ≤ -6”? Actually, in the image, it might be written as “x ≥ 2 or x ≤ -6”, but let’s assume it’s there.

Actually, upon closer inspection of the original image description, from 4|x+2|≥16, the path going down-left is labeled:

> x ≥ 2 or x ≤ -6

Yes — so follow that to the next box:
|x + 6| - 7 < 4

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Step 6: Solve |x + 6| - 7 < 4

Add 7 to both sides:

|x + 6| < 11

This means:
-11 < x + 6 < 11

Subtract 6:

-17 < x < 5

So: -17 < x < 5

Now, look at paths from this box.

One path says:
> -17 < x < 5

Exactly! Follow that path (down-left) to the next box:
|18 - 3x| ≤ 6

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Step 7: Solve |18 - 3x| ≤ 6

This means:
-6 ≤ 18 - 3x ≤ 6

Now solve the compound inequality.

First, subtract 18 from all parts:

-6 - 18 ≤ -3x ≤ 6 - 18
→ -24 ≤ -3x ≤ -12

Now divide by -3 — and remember to flip the inequality signs!

Dividing by negative:
(-24)/(-3) ≥ x ≥ (-12)/(-3)
→ 8 ≥ x ≥ 4

Or rewritten: 4 ≤ x ≤ 8

Now, look at paths from |18 - 3x| ≤ 6.

One path says:
> 4 ≤ x ≤ 8

Perfect! Follow that path (to the right) — and guess what? It leads to the “Finished!” box!

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✔ We’ve reached the end!

Final path taken:

Start → |x| > 8 → x > 8 or x < -8 → |x| ≤ 2 → -2 ≤ x ≤ 2 → |8x| < 24 → -3 < x < 3 → -7|x - 8| < -56 → x > 16 or x < 0 → 4|x + 2| ≥ 16 → x ≥ 2 or x ≤ -6 → |x + 6| - 7 < 4 → -17 < x < 5 → |18 - 3x| ≤ 6 → 4 ≤ x ≤ 8 → FINISHED!

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Final Answer:
The path through the maze ends at the "Finished!" box after solving |18 - 3x| ≤ 6, which gives 4 ≤ x ≤ 8.