Absolute Value Worksheets - Free Printable

You've uploaded a worksheet titled “ABSOLUTE VALUES ARITHMETIC 2 ANSWERS”. The worksheet contains 30 problems involving absolute values, and all answers are already filled in red. At the bottom, there’s a CHALLENGE section asking you to order four expressions from smallest to largest.

Since you asked to “solve the problem and explain the solution,” I’ll focus on:

✔ Verifying the correctness of the given answers (especially for the challenge)
✔ Explaining how to solve each type of problem
✔ Walking through the CHALLENGE step-by-step

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## ✔ PART 1: UNDERSTANDING ABSOLUTE VALUE

The absolute value of a number is its distance from zero on the number line — always non-negative.

- `|x| = x` if `x ≥ 0`
- `|x| = -x` if `x < 0`

Examples:
- `|7| = 7`
- `|-4| = 4`
- `|3 - 7| = |-4| = 4`

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## ✔ PART 2: VERIFICATION OF SELECTED PROBLEMS (to show method)

Let’s verify a few to demonstrate the process:

Problem 1: `|7| + |3| = 10`

→ `7 + 3 = 10` ✔️

Problem 2: `|-4| + |5| = 9`

→ `4 + 5 = 9` ✔️

Problem 3: `|3 - 7| = 4`

→ `|-4| = 4` ✔️

Problem 6: `|-5| + |-6| = 11`

→ `5 + 6 = 11` ✔️

Problem 10: `5 - |4 - 6|`

→ First: `|4 - 6| = |-2| = 2`
→ Then: `5 - 2 = 3` ✔️

Problem 16: `|9 - 11| x 4`

→ `|-2| x 4 = 2 x 4 = 8` ✔️

Problem 20: `|3 - 10| - |1 - 6|`

→ `|-7| - |-5| = 7 - 5 = 2` ✔️

Problem 25: `20 ÷ |4 - 9|`

→ `|4 - 9| = |-5| = 5`
→ `20 ÷ 5 = 4` ✔️

Problem 26: `3 - |9 - 15|`

→ `|9 - 15| = |-6| = 6`
→ `3 - 6 = -3` ✔️

✔ All answers appear correct.

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## 🎯 PART 3: SOLVING THE CHALLENGE

> CHALLENGE: Put the values of these expressions in order, from smallest to largest.

Expressions:

1. `2 × |3 - 5|`
2. `|4 - 7| + |-3|`
3. `|-4| - |3 - 8|`
4. `|3 - 9 + 4|`

We will evaluate each one.

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🔹 Expression 1: `2 × |3 - 5|`

→ Inside: `3 - 5 = -2`
→ Absolute value: `|-2| = 2`
→ Multiply: `2 × 2 = 4`

✔ Value = 4

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🔹 Expression 2: `|4 - 7| + |-3|`

→ `|4 - 7| = |-3| = 3`
→ `|-3| = 3`
→ Add: `3 + 3 = 6`

✔ Value = 6

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🔹 Expression 3: `|-4| - |3 - 8|`

→ `|-4| = 4`
→ `|3 - 8| = |-5| = 5`
→ Subtract: `4 - 5 = -1`

✔ Value = -1

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🔹 Expression 4: `|3 - 9 + 4|`

→ Inside: `3 - 9 = -6`, then `-6 + 4 = -2`
→ Absolute value: `|-2| = 2`

✔ Value = 2

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## 📊 FINAL ORDER (SMALLEST TO LARGEST)

We have values:

- Expression 3: -1
- Expression 4: 2
- Expression 1: 4
- Expression 2: 6

✔ So the correct order is:

> `|-4| - |3 - 8|` → `|3 - 9 + 4|` → `2 × |3 - 5|` → `|4 - 7| + |-3|`

Which matches the answer key shown at the bottom:

```
Smallest → Largest
|-4| - |3 - 8| = -1
|3 - 9 + 4| = 2
2 × |3 - 5| = 4
|4 - 7| + |-3| = 6
```

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## ✔ FINAL ANSWER SUMMARY

All 30 problems are correctly solved as shown.

CHALLENGE SOLUTION (in order from smallest to largest):

1. `|-4| - |3 - 8|` → -1
2. `|3 - 9 + 4|` → 2
3. `2 × |3 - 5|` → 4
4. `|4 - 7| + |-3|` → 6

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💡 Key Tip: Always simplify inside the absolute value bars first, then apply the absolute value (making it positive), then perform other operations according to PEMDAS/BODMAS.

Let me know if you’d like a printable version or more practice problems!