9th Grade Math Worksheets | Worksheets Worksheets - Free Printable

The image shows a 9th-grade math worksheet with several problems. Below, I will solve each problem step by step and explain the solution.

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Problem 1: Find \( x \). Round to the nearest tenth.

The diagram shows a right triangle with one angle labeled \( x \), the opposite side labeled 8, and the adjacent side labeled 15. We need to find the angle \( x \).

#### Solution:
1. Use the tangent function, which relates the opposite side to the adjacent side in a right triangle:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{15}
\]
2. To find \( x \), take the inverse tangent (arctangent) of both sides:
\[
x = \arctan\left(\frac{8}{15}\right)
\]
3. Use a calculator to compute \( \arctan\left(\frac{8}{15}\right) \):
\[
x \approx 28.07^\circ
\]
4. Round to the nearest tenth:
\[
x \approx 28.1^\circ
\]

Answer:
\[
\boxed{28.1}
\]

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Problem 2: Solve.

Solve the inequality:
\[
-4x - 5x < 18
\]

#### Solution:
1. Combine like terms on the left-hand side:
\[
-4x - 5x = -9x
\]
So the inequality becomes:
\[
-9x < 18
\]
2. Divide both sides by \(-9\). Remember that dividing by a negative number reverses the inequality sign:
\[
x > \frac{18}{-9}
\]
3. Simplify the fraction:
\[
x > -2
\]

Answer:
\[
\boxed{x > -2}
\]

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Problem 3: Simplify. Write each answer in scientific notation.

Simplify:
\[
(2.53 \times 10^6) \cdot (4 \times 10^{-3})
\]

#### Solution:
1. Multiply the coefficients (the numbers before the powers of 10):
\[
2.53 \cdot 4 = 10.12
\]
2. Add the exponents of 10 (since we are multiplying powers of 10):
\[
10^6 \cdot 10^{-3} = 10^{6 + (-3)} = 10^3
\]
3. Combine the results:
\[
10.12 \times 10^3
\]
4. Convert to proper scientific notation. In scientific notation, the coefficient must be between 1 and 10. Currently, 10.12 is not in this range, so rewrite it as:
\[
10.12 = 1.012 \times 10^1
\]
Therefore:
\[
10.12 \times 10^3 = (1.012 \times 10^1) \times 10^3 = 1.012 \times 10^{1+3} = 1.012 \times 10^4
\]

Answer:
\[
\boxed{1.012 \times 10^4}
\]

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Problem 4: Evaluate each function at the given value.

Evaluate the function \( f(x) = 10x^2 \) at \( x = -4 \).

#### Solution:
1. Substitute \( x = -4 \) into the function:
\[
f(-4) = 10(-4)^2
\]
2. Compute \( (-4)^2 \):
\[
(-4)^2 = 16
\]
3. Multiply by 10:
\[
f(-4) = 10 \cdot 16 = 160
\]

Answer:
\[
\boxed{160}
\]

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Problem 5: Simplify the following expression.

Simplify:
\[
(2x^3 - x^2) - (3x^3 - 2x)
\]

#### Solution:
1. Distribute the negative sign through the second set of parentheses:
\[
(2x^3 - x^2) - (3x^3 - 2x) = 2x^3 - x^2 - 3x^3 + 2x
\]
2. Combine like terms:
- For \( x^3 \)-terms: \( 2x^3 - 3x^3 = -x^3 \)
- For \( x^2 \)-terms: \( -x^2 \) (no other \( x^2 \)-terms)
- For \( x \)-terms: \( 2x \) (no other \( x \)-terms)
3. Write the simplified expression:
\[
-x^3 - x^2 + 2x
\]

Answer:
\[
\boxed{-x^3 - x^2 + 2x}
\]

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Final Answers:

1. \(\boxed{28.1}\)
2. \(\boxed{x > -2}\)
3. \(\boxed{1.012 \times 10^4}\)
4. \(\boxed{160}\)
5. \(\boxed{-x^3 - x^2 + 2x}\)