Here are the step-by-step solutions for each problem on the worksheet.
1. Find $x$. Round to the nearest tenth.
*
Identify the sides: The angle given is $25^\circ$. The side opposite this angle is $9$. The side adjacent (next to) the angle is $x$.
*
Choose the ratio: Since we have Opposite and Adjacent, we use Tangent ($\tan$).
$$ \tan(\text{angle}) = \frac{\text{Opposite}}{\text{Adjacent}} $$
*
Set up the equation:
$$ \tan(25^\circ) = \frac{9}{x} $$
*
Solve for $x$: Multiply both sides by $x$, then divide by $\tan(25^\circ)$.
$$ x = \frac{9}{\tan(25^\circ)} $$
*
Calculate:
$$ x \approx \frac{9}{0.4663} \approx 19.299... $$
*
Round: Rounding to the nearest tenth gives
19.3.
2. Solve: $-4x - 5x < 18$
*
Combine like terms: Combine $-4x$ and $-5x$.
$$ -9x < 18 $$
*
Isolate $x$: Divide both sides by $-9$.
* *Important Rule:* When you divide or multiply an inequality by a negative number, you must flip the inequality sign (from $<$ to $>$).
$$ x > \frac{18}{-9} $$
$$ x > -2 $$
3. Simplify. Write each answer in scientific notation.
Expression: $(2.5 \times 10^{-4}) \cdot (4 \times 10^2)$
*
Group the numbers and the powers of 10:
$$ (2.5 \cdot 4) \times (10^{-4} \cdot 10^2) $$
*
Multiply the coefficients:
$$ 2.5 \cdot 4 = 10 $$
*
Multiply the powers of 10: Add the exponents.
$$ 10^{-4 + 2} = 10^{-2} $$
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Combine them:
$$ 10 \times 10^{-2} $$
*
Convert to proper scientific notation: Scientific notation requires the first number to be between 1 and 10. We can write $10$ as $1 \times 10^1$.
$$ (1 \times 10^1) \times 10^{-2} = 1 \times 10^{1 + (-2)} = 1 \times 10^{-1} $$
4. Evaluate each function at the given value.
Function: $f(x) = 10 \times 2^{-x}$ at $x = -4$
*
Substitute $-4$ for $x$:
$$ f(-4) = 10 \times 2^{-(-4)} $$
*
Simplify the exponent: A negative times a negative is a positive.
$$ f(-4) = 10 \times 2^4 $$
*
Calculate the power: $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$.
$$ f(-4) = 10 \times 16 $$
*
Multiply:
$$ 160 $$
5. Simplify the following expression.
Expression: $(2x^2 - x^3) - (3x^3 - 2x^2) + 2x^3$
*
Distribute the negative sign: Apply the minus sign to the terms inside the second set of parentheses.
$$ 2x^2 - x^3 - 3x^3 + 2x^2 + 2x^3 $$
*
Group like terms:
* $x^3$ terms: $-x^3 - 3x^3 + 2x^3$
* $x^2$ terms: $2x^2 + 2x^2$
*
Combine coefficients:
* For $x^3$: $-1 - 3 + 2 = -2$. So, $-2x^3$.
* For $x^2$: $2 + 2 = 4$. So, $4x^2$.
*
Final Expression:
$$ -2x^3 + 4x^2 $$
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Final Answer:
1. 19.3
2. x > -2
3. 1 × 10⁻¹
4. 160
5. -2x³ + 4x²
Parent Tip: Review the logic above to help your child master the concept of 8th 9th grade worksheet.
