Practice Keystone Exam for Algebra 1 with multiple-choice questions.

Let's solve each problem from the Algebra 1 Practice Keystone Exam step by step.

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Problem 1: Which of the following inequalities is true for ALL real values of x?

We are given four choices. We need to determine which inequality holds for all real numbers $ x $.

#### a. $ x^3 \geq x^2 $

Let’s test with $ x = -1 $:
- $ (-1)^3 = -1 $
- $ (-1)^2 = 1 $
- Is $ -1 \geq 1 $? No → False

So this is not always true.

#### b. $ 3x^3 \geq 2x^3 $

Subtract $ 2x^3 $ from both sides:
- $ x^3 \geq 0 $

This is only true when $ x \geq 0 $. For $ x < 0 $, it fails (e.g., $ x = -1 $: $ -3 \geq -2 $? No). So not always true.

#### c. $ (2x)^2 \geq 3x^2 $

Simplify left side:
- $ (2x)^2 = 4x^2 $
- So: $ 4x^2 \geq 3x^2 $
- Subtract $ 3x^2 $: $ x^2 \geq 0 $

This is true for all real numbers, because $ x^2 \geq 0 $ always.

✔ This one is always true.

#### d. $ 3(x - 2)^2 \geq 3x^2 - 2 $

Let’s test $ x = 0 $:
- Left: $ 3(0 - 2)^2 = 3(4) = 12 $
- Right: $ 3(0)^2 - 2 = -2 $
- $ 12 \geq -2 $ → True

Now try $ x = 1 $:
- Left: $ 3(-1)^2 = 3 $
- Right: $ 3(1) - 2 = 1 $
- $ 3 \geq 1 $ → True

Try $ x = 3 $:
- Left: $ 3(1)^2 = 3 $
- Right: $ 3(9) - 2 = 27 - 2 = 25 $
- $ 3 \geq 25 $? No → False

So not always true.

✔ Answer: c. $ (2x)^2 \geq 3x^2 $

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Problem 2: An expression is shown: $ 2\sqrt{51x} $. Which value of x makes it equivalent to $ 10\sqrt{51} $?

We want:

$$
2\sqrt{51x} = 10\sqrt{51}
$$

Divide both sides by 2:

$$
\sqrt{51x} = 5\sqrt{51}
$$

Square both sides:

$$
51x = 25 \cdot 51
$$

Divide both sides by 51:

$$
x = 25
$$

✔ Answer: b. 25

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Problem 3: Two monomials: $ 450x^2y^3 $ and $ 3,000x^4y^1 $. What is the LCM?

To find the LCM of two monomials, we take:
- The LCM of coefficients
- The highest power of each variable

#### Step 1: Coefficients: 450 and 3000

Factor them:
- $ 450 = 45 \times 10 = (9 \times 5)(2 \times 5) = 2 \cdot 3^2 \cdot 5^2 $
- $ 3000 = 3 \times 1000 = 3 \cdot 10^3 = 3 \cdot (2 \cdot 5)^3 = 2^3 \cdot 3 \cdot 5^3 $

LCM takes highest powers:
- $ 2^3 $
- $ 3^2 $
- $ 5^3 $

So:
$$
LCM = 8 \cdot 9 \cdot 125 = 72 \cdot 125 = 9000
$$

Check: $ 8 \cdot 125 = 1000 $, $ 1000 \cdot 9 = 9000 $

#### Step 2: Variables
- $ x^2 $ vs $ x^4 $ → take $ x^4 $
- $ y^3 $ vs $ y^1 $ → take $ y^3 $

So LCM monomial is:
$$
9000x^4y^3
$$

But look at the options:

a. $ 2xy $
b. $ 50xy $
c. $ 150x^2y^3 $
d. $ 9,000x^4y^3 $

✔ Answer: d. $ 9,000x^4y^3 $

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Problem 4: Simplify: $ 2(2\sqrt{4})^2 $

Step-by-step:

First, simplify inside:
- $ \sqrt{4} = 2 $
- So $ 2\sqrt{4} = 2 \cdot 2 = 4 $
- Now: $ (2\sqrt{4})^2 = 4^2 = 16 $
- Then multiply by 2: $ 2 \cdot 16 = 32 $

Wait — let’s double-check the expression:

It says: $ 2(2\sqrt{4})^2 $

So:
- $ 2\sqrt{4} = 2 \cdot 2 = 4 $
- $ (4)^2 = 16 $
- $ 2 \cdot 16 = 32 $

✔ Answer: d. 32

Alternatively, you could write:
$$
2(2\sqrt{4})^2 = 2 \cdot (2^2 \cdot (\sqrt{4})^2) = 2 \cdot (4 \cdot 4) = 2 \cdot 16 = 32
$$

Same result.

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✔ Final Answers:

1. c. $ (2x)^2 \geq 3x^2 $
2. b. 25
3. d. $ 9,000x^4y^3 $
4. d. 32

Let me know if you'd like explanations in a different format or need help with more problems!