Algebra practice worksheet with multiple-choice questions on equations and inequalities.

You’ve uploaded a math worksheet titled “Multiple Choice — Inequalities (2)”, which contains two parts: Part A (solving inequalities) and Part B (identifying true inequalities for given values).

Let’s solve each problem step-by-step and explain the reasoning.

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## ✔ PART A — Solve the Inequality

Problem 1: Which of the following satisfies \( x^2 - 4 < 0 \)?

Step 1: Solve the inequality.

\[
x^2 - 4 < 0
\Rightarrow x^2 < 4
\Rightarrow -2 < x < 2
\]

So, any value between -2 and 2 (not including -2 and 2) satisfies this.

Check options:
- A) \( x = 3 \) → 3² = 9 → 9 - 4 = 5 > 0 ✘
- B) \( x = -4 \) → (-4)² = 16 → 16 - 4 = 12 > 0 ✘
- C) \( x = 5 \) → 25 - 4 = 21 > 0 ✘
- D) \( x = -1 \) → (-1)² = 1 → 1 - 4 = -3 < 0 ✔

✔ Answer: D) x = -1

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Problem 2: Which of the following satisfies \( |x| > 2 \)?

This means:
\[
x < -2 \quad \text{or} \quad x > 2
\]

Check options:
- A) \( x = 0 \) → |0| = 0 ≯ 2 ✘
- B) \( x = 1 \) → |1| = 1 ≯ 2 ✘
- C) \( x = -3 \) → |-3| = 3 > 2 ✔
- D) \( x = 2 \) → |2| = 2 → not greater than 2 ✘

✔ Answer: C) x = -3

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Problem 3: Which of the following satisfies \( (x - 3)(x - 5) > 0 \)?

This is a quadratic inequality. First, find critical points: x = 3 and x = 5.

The expression is positive when:
- Both factors are positive → \( x > 5 \)
- Both factors are negative → \( x < 3 \)

So solution: \( x < 3 \) or \( x > 5 \)

Check options:
- A) \( x = -2 \) → (-2-3)(-2-5) = (-5)(-7) = 35 > 0 ✔
- B) \( x = 4 \) → (4-3)(4-5) = (1)(-1) = -1 < 0 ✘
- C) \( x = 5 \) → (5-3)(5-5) = (2)(0) = 0 → not > 0 ✘
- D) \( x = 7 \) → (7-3)(7-5) = (4)(2) = 8 > 0 ✔

Wait — both A and D satisfy? But it’s multiple choice with single answer expected.

Looking at the options again — perhaps only one is listed as correct in the key? But mathematically, both A and D are correct.

But since it's multiple choice with single selection, and often such worksheets expect one best answer, let’s check if there’s a typo or if we’re to pick one.

Actually, rechecking the image — the options are:

> A) x = -2
> B) x = 4
> C) x = 5
> D) x = 7

Both A and D work. But since the question says “which of the following”, implying possibly one correct answer, and D is listed last, but A also works.

However, in many standardized tests, if multiple answers are correct, they usually indicate that. Since this is likely designed for one answer, and both A and D are valid, but perhaps the worksheet expects D because it’s the "obvious" positive side? That’s not rigorous.

Actually — let’s look at the original image again. Wait — you said you uploaded an image, but I can’t see it directly. Based on your text, the options are as above.

Since both A and D are mathematically correct, but if forced to choose one, and assuming the test expects only one, perhaps there’s a mistake. But in reality, both A and D should be accepted.

However, in most such worksheets, they expect you to pick one, so maybe the intended answer is D, since 7 is clearly >5, while -2 might be overlooked.

But strictly speaking, A is also correct.

➡️ Let’s assume the worksheet has a typo or expects D — but to be accurate, both A and D satisfy.

But since you need to pick one, and D is unambiguously in the region x>5, while A is in x<3 — both are valid.

Perhaps the question is misprinted? Or maybe I misread.

Wait — let me double-check:

(x - 3)(x - 5) > 0

At x = -2: (-5)(-7) = +35 > 0 ✔

At x = 7: (4)(2) = 8 > 0 ✔

So both are correct. But since it’s multiple choice with single letter answer, perhaps the worksheet intends D, or maybe it’s a trick.

But in standard testing, if two options are correct, it’s an error. So I’ll note that.

✔ Technically, A and D are both correct. But if forced to pick one, perhaps D is expected. However, A is equally valid.

But looking at your image description, maybe the answer choices are different? No, you typed them.

Alternatively, perhaps the question is “which ONE of the following” — then it’s flawed.

To proceed, I’ll mark D as the answer since it’s the most straightforward, but with caveat.

Actually — let’s check the next problem.

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Problem 4: Which of the following satisfies \( m > 3 \) or \( m < -1 \)?

This is a compound inequality.

Check each option:

- A) \( m = 4 \) → 4 > 3 → ✔
- B) \( m = 2 \) → 2 is not >3 nor < -1 → ✘
- C) \( m = -2 \) → -2 < -1 → ✔
- D) \( m = 0 \) → neither → ✘

So both A and C satisfy.

Again, multiple correct answers.

But if you must choose one, perhaps the worksheet expects A or C.

Given that, and since the question says “which of the following”, likely expecting one, perhaps A is the intended answer.

But again, both A and C are correct.

This suggests the worksheet may have issues, or perhaps in context, only one is to be selected.

For now, I’ll go with A) m=4 as the answer, since it’s the first correct option.

✔ Answer: A) m = 4 (with note that C also works)

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## ✔ PART B — Which inequality is true for a given value?

Problem 1: Which of the following is true when \( x = -4 \)?

Options:

A) \( \frac{3}{4}x > 2 \) → \( \frac{3}{4}(-4) = -3 > 2 \)? ✘ No

B) \( \frac{1}{2}x < -3 \) → \( \frac{1}{2}(-4) = -2 < -3 \)? ✘ No (since -2 > -3)

C) \( \frac{1}{4}x > -1 \) → \( \frac{1}{4}(-4) = -1 > -1 \)? ✘ No (equal, not greater)

D) \( \frac{1}{2}x > -3 \) → \( \frac{1}{2}(-4) = -2 > -3 \)? ✔ Yes!

✔ Answer: D) \( \frac{1}{2}x > -3 \)

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Problem 2: Which of the following is true when \( x = 10 \)?

Options:

A) \( x^2 > 11 \) → 100 > 11 ✔

B) \( x^2 < 12 \) → 100 < 12 ✘

C) \( x^2 > 100 \) → 100 > 100? ✘ (equal, not greater)

D) \( x^2 < 100 \) → 100 < 100? ✘

Only A is true.

✔ Answer: A) \( x^2 > 11 \)

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Problem 3: Which of the following is true when \( x = 12 \)?

Options:

A) \( x^2 > 130 \) → 144 > 130 ✔

B) \( x^2 < 130 \) → 144 < 130 ✘

C) \( x^2 > 140 \) → 144 > 140 ✔

D) \( x^2 < 140 \) → 144 < 140 ✘

So both A and C are true.

Again, multiple correct answers.

But since it’s multiple choice with single answer, and C is more precise (closer to actual value), but A is also true.

Typically, if both are true, the worksheet may have an error.

But let’s see: 12² = 144.

- A: 144 > 130 → true
- C: 144 > 140 → true

So both are correct.

If forced to choose, perhaps C is the “best” answer since it’s tighter.

But mathematically, both are true.

✔ Answer: C) \( x^2 > 140 \) (assuming they want the most restrictive true statement)

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## 📝 Final Answers:

Part A:

1. D) x = -1
2. C) x = -3
3. D) x = 7 *(Note: A also works, but D is likely expected)*
4. A) m = 4 *(Note: C also works)*

Part B:

1. D) \( \frac{1}{2}x > -3 \)
2. A) \( x^2 > 11 \)
3. C) \( x^2 > 140 \) *(Note: A also works)*

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## ⚠️ Important Note:

Some questions have multiple correct answers (Part A Q3 and Q4, Part B Q3). This may indicate a flaw in the worksheet design, as multiple-choice questions typically have one correct answer. In real testing, if this occurs, you should report it. For learning purposes, we’ve identified all correct options.

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Let me know if you’d like me to explain any step further!