Explore detailed solutions and clear, step-by-step explanations for the intermediate algebra questions found in Sample Set 1. This guide is designed to help students deepen their understanding and master key algebra concepts.
Write 230,000,000,000 in scientific notation.
Solution
Write the number in the form \( a \times 10^n \), where \( 1 \leq |a| \lt 10 \) and \( n \in \mathbb{Z} \).
\( 230{,}000{,}000{,}000 = 2.3 \times 10^{11} \)
Evaluate: \( 30 - 12 \div 3 \times 2 \)
Solution
First, perform division and multiplication left to right:
\( 12 \div 3 \times 2 = 4 \times 2 = 8 \)
Then: \( 30 - 8 = 22 \)
Evaluate: \( |4 - 8(3 - 12)| - |5 - 11| \)
Solution
Start with the inner parentheses:
\( 3 - 12 = -9 \)
Then: \( 8 \times (-9) = -72 \), so:
\( |4 - (-72)| = |76| = 76 \)
\( |5 - 11| = |-6| = 6 \)
Final result: \( 76 - 6 = 70 \)
Evaluate: \( -18 + 4(6 \div 2)^2 \)
Solution
\( 6 \div 2 = 3 \), then square: \( 3^2 = 9 \)
\( 4 \times 9 = 36 \)
\( -18 + 36 = 18 \)
Evaluate: \( 11 + \sqrt{-4 + 6 \times 4 \div 3} \)
Solution
\( 6 \times 4 = 24 \), then \( 24 \div 3 = 8 \)
\( -4 + 8 = 4 \), so \( \sqrt{4} = 2 \)
\( 11 + 2 = 13 \)
Simplify: \( 12x^3 - 3(2x^3 + 4x -1) - 5x + 7 \)
Solution
Distribute the -3:
\( 12x^3 - 6x^3 - 12x + 3 - 5x + 7 \)
Combine like terms:
\( 6x^3 - 17x + 10 \)
Simplify: \( \left( \dfrac{x^4}{x^3} \right)^3 \)
Solution
\( \dfrac{x^4}{x^3} = x^{4-3} = x \), so:
\( \left( \dfrac{x^4}{x^3} \right)^3 = x^3 \)
Simplify: \( \dfrac{(3x^2y^{-2})^3}{(9xy^3)^3} \)
Solution
\( (3x^2y^{-2})^3 = 27x^6y^{-6} \)
\( (9xy^3)^3 = 729x^3y^9 \)
So: \( \dfrac{(3x^2y^{-2})^3}{(9xy^3)^3} = \dfrac{27x^6y^{-6}}{729x^3y^9} = \dfrac{1}{27} \cdot x^{6-3} \cdot y^{-6-9} = \dfrac{x^3}{27y^{15}} \)
Simplify: \( \dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3} \)
Solution
Note: \( (x^3 + y)^0 = 1 \)
Numerator: \( (2x^{-3}y^4)^3 = 8x^{-9}y^{12} \)
Denominator: \( (4xy^{-2})^3 = 64x^3y^{-6} \)
Now: \( \dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3} = \dfrac{8x^{-9}y^{12}}{64x^3y^{-6}} = \dfrac{1}{8} \cdot x^{-12} \cdot y^{18} = \dfrac{1}{8} \cdot \dfrac{y^{18}}{x^{12}} \)
Write as an inequality: "9 is less than the product of M and N"
Solution
\( 9 < M \cdot N \)
Slope of the line perpendicular to \( y = \frac{1}{3}x - 7 \)
Solution
Perpendicular slope is the negative reciprocal of \( \frac{1}{3} \):
\( m = -3 \)
Write equation of a line with slope -3 and y-intercept (0, -5)
Solution
Using \( y = mx + b \):
\( y = -3x - 5 \)
Solve: \( -5x + 20 = 25 \)
Solution
Subtract 20: \( -5x = 5 \)
Divide by -5: \( x = -1 \)
Solve: \( -3x + 4 < -8 \)
Solution
Subtract 4: \( -3x < -12 \)
Divide by -3 and flip inequality: \( x > 4 \)
Solve: \( 2x^2 - 32 = 0 \)
Solution
Add 32: \( 2x^2 = 32 \)
Divide by 2: \( x^2 = 16 \)
Take square root: \( x = \pm 4 \)
Solve: \( -0.25x + 1.3 = -0.55x - 0.2 \)
Solution
Add \( 0.55x \) to both sides:
\( 0.3x + 1.3 = -0.2 \)
Subtract 1.3: \( 0.3x = -1.5 \)
Divide: \( x = -5 \)
Solve: \( -0.25x^2 + 1.5 = -10.75 \)
Solution
Subtract 1.5: \( -0.25x^2 = -12.25 \)
Divide: \( x^2 = 49 \), so \( x = \pm 7 \)
What is the slope of a line perpendicular to \( x = -3 \)?
Solution
A vertical line (\( x = \text{constant} \)) is perpendicular to a horizontal line.
The slope of a horizontal line is 0.
What is the slope of a line parallel to \( x = 5 \)?
Solution
Lines parallel to vertical lines are also vertical.
Slope is undefined.
What is the slope of a line perpendicular to \( y = 6 \)?
Solution
A horizontal line (\( y = \text{constant} \)) is perpendicular to a vertical line.
The slope of a vertical line is undefined.