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High School Math : Intermediate Single-Variable Algebra

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Example Questions

Example Question #1 : Understanding Rational Expressions

Solve:

If varies directly as , and x=7 when y=21 , find when y=-5 .

Possible Answers:

$x=\frac{-1}{3}$

$x=\frac{-7}{3}$

$x=\frac{-11}{3}$

$x=\frac{-13}{3}$

$x=\frac{-5}{3}$

Correct answer:

$x=\frac{-5}{3}$

Explanation:

The formula for a direct variation is:

$\frac{x_1}{x_2}=\frac{y_1}{y_2}$

Plugging in our values, we get:

$\frac{7}{x}=\frac{21}{-5}$

21x=-35

$x= \frac{-5}{3}$

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Example Question #2 : Understanding Rational Expressions

If two boxes have the same depth and capacity, the length is inversely proportional to the width. One box is $60\ cm$ long and $40\ cm$ wide. A second box (same depth and capacity) is $5\ cm$ long. How wide is it?

Possible Answers:

$460\ cm$

$480\ cm$

$470\ cm$

$450\ cm$

$490\ cm$

Correct answer:

$480\ cm$

Explanation:

The formula for an indirect variation is:

$\frac{x_1}{x_2}=\frac{y_2}{y_1}$

Plugging in our values, we get:

$\frac{60\ cm}{5\ cm}=\frac{y}{40\ cm}$

$5y=2400\ cm$

$y=480\ cm$

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Example Question #1 : Rational Expressions

Simplify $\frac{x+2}{x-1}+\frac{x-5}{x+3}$

Possible Answers:

$\frac{2x^{2}-x+11}{x^{2}+2x-3}$

$\frac{4x^{2}-3x+5}{x^{2}+3x-2}$

$\frac{3x^{2}+5x-6}{x^{2}-3x+2}$

$\frac{2x^{2}-5x+6}{x^{2}-2x+3}$

$\frac{3x^{2}-x+7}{x^{2}+2x+2}$

Correct answer:

$\frac{2x^{2}-x+11}{x^{2}+2x-3}$

Explanation:

This is a more complicated form of $\frac{1}{2}+\frac{1}{3}= \frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators. Simplify as needed.

$\frac{x+2}{x-1}+\frac{x-5}{x+3}=\frac{x+2}{x-1}\cdot \frac{x+3}{x+3}+\frac{x-5}{x+3}\cdot \frac{x-1}{x-1}$

which is equivalent to $\frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get $\frac{2x^{2}-x+11}{x^{2}+2x-3}$

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Example Question #2 : Simplifying Rational Expressions

Divide and simplify the following rational expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

Possible Answers:

x+2

x-2

x+1

x-3

x-1

Correct answer:

x-2

Explanation:

Multiply by the reciprocal of the second expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

$\frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}$

Factor the expressions:

$\frac{(x+5)(x+2)}{x+2}\cdot \frac{(x-3)(x-2)}{(x-3)(x+5)}$

Remove common terms:

$\frac{\mathbf{(x+5)(x+2)}}{\mathbf{x+2}}\cdot \frac{\mathbf{(x-3)}(x-2)}{\mathbf{(x-3)(x+5)}}$

x-2

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Example Question #1 : Simplifying Rational Expressions

Add and simplify the following rational expression:

$3n+1+\frac{1}{3n-1}$

Possible Answers:

$\frac{9n^2}{3n-1}$

$\frac{6n^2}{3n-1}$

$\frac{3n^2}{2n-1}$

$\frac{3n^2}{3n-1}$

$\frac{6n^2}{2n-1}$

Correct answer:

$\frac{9n^2}{3n-1}$

Explanation:

Begin by multiplying the left term by $\frac{3n-1}{3n-1}$ :

$3n+1+\frac{1}{3n-1}$

$3n+1 (\frac{3n-1}{3n-1})+\frac{1}{3n-1}$

Simplify:

$\frac{(3n+1)(3n-1)+1}{3n-1}$

$\frac{9n^2-1+1}{3n-1}$

$\frac{9n^2}{3n-1}$

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Example Question #1 : Simplifying Rational Expressions

Simplify the following rational expression:

$\frac{\frac{x+y}{x}}{\frac{1}{x}+\frac{1}{y}}$

Possible Answers:

$\frac{x}{y}$

$\frac{y}{x}$

$\frac{1}{x}$

Correct answer:

Explanation:

Begin by combining the terms in the denominator:

$\frac{\frac{x+y}{x}}{\frac{1}{x}+\frac{1}{y}}$

$\frac{\frac{x+y}{x}}{\frac{x+y}{xy}}$

Multiply by the reciprocal of the denominator:

$\frac{x+y}{x} \cdot \frac{xy}{x+y}$

Remove like terms:

$\frac{\mathbf{x+y}}{\mathbf{x}} \cdot \frac{\boldsymbol{x}y}{\mathbf{x+y}}$

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Example Question #3 : Simplifying Rational Expressions

Simplify the following rational expression:

$\frac{n+5+\frac{3}{n+1}}{n-1-\frac{3}{n+1}}$

Possible Answers:

$\frac{n+4}{n-2}$

$\frac{n+3}{n-1}$

$\frac{n+4}{n-1}$

$\frac{n+3}{n-2}$

Correct answer:

$\frac{n+4}{n-2}$

Explanation:

Create a common denominator of (n+1) in both the numerator and denominator:

$\frac{n+5+\frac{3}{n+1}}{n-1-\frac{3}{n+1}}$

$\frac{\frac{(n+5)(n+1)+3}{n+1}}{\frac{(n-1)(n+1)-3}{n+1}}$

Multiply by the reciprocal of the denominator:

$\frac{(n+5)(n+1)+3}{n+1}\cdot \frac{n+1}{(n-1)(n+1)-3}$

Simplify:

$\frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}$

$\frac{(n+4)(n+2)}{n+1}\cdot \frac{n+1}{(n-2)(n+2)}$

Remove common terms:

$\frac{(n+4)\mathbf{(n+2)}}{\mathbf{n+1}}\cdot \frac{\mathbf{n+1}}{(n-2)(\mathbf{n+2})}$

$\frac{n+4}{n-2}$

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Example Question #4 : Simplifying Rational Expressions

Multiply and simplify the following rational expression:

$\frac{a^3-b^3}{b^2-a^2} \cdot \frac{a+b}{a^2+ab+b^2}$

Possible Answers:

Correct answer:

Explanation:

Factor the expression:

$\frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}$

Remove like terms:

$\frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}$

$\frac{a-b}{b-a}=\frac{-1(b-a)}{b-a}=-1$

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Example Question #5 : Simplifying Rational Expressions

Divide and simplify the following rational expression:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

Possible Answers:

$\frac{x-3}{x-4}$

$\frac{x-5}{x-4}$

$\frac{x-4}{x-5}$

$\frac{x-4}{x-3}$

$\frac{x-3}{x-5}$

Correct answer:

$\frac{x-3}{x-4}$

Explanation:

Multiply by the inverse of the denominator:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

$\frac{x^2-11x+24}{x^2-18x+80}\cdot \frac{x^2-15x+50}{x^2-9x+20}$

Factor:

$\frac{(x-8)(x-3)}{(x-8)(x-10)}\cdot \frac{(x-5)(x-10)}{(x-5)(x-4)}$

Remove like terms:

$\frac{(\mathbf{x-8})(x-3)}{(\mathbf{x-8})(\mathbf{x-10})}\cdot \frac{(\mathbf{x-5})(\mathbf{x-10})}{(\mathbf{x-5})(x-4)}$

$\frac{x-3}{x-4}$

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Example Question #1 : Solving Rational Equations And Inequalities

Solve the following rational expression:

$\frac{y}{y-3}+\frac{6}{y+3}=1$

Possible Answers:

$y=\frac{1}{2}$

y=2

y=1

$y=\frac{1}{3}$

y=3

Correct answer:

y=1

Explanation:

Multiply the equation by (y-3)(y+3) :

$\frac{y}{y-3}+\frac{6}{y+3}=1$

y(y+3)+6(y-3)=1(y-3)(y+3)

y^2+3y+6y-18 = y^2-9

9y=9

y=1

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High School Math : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Rational Expressions

Example Question #2 : Understanding Rational Expressions

Example Question #1 : Rational Expressions

Example Question #2 : Simplifying Rational Expressions

Example Question #1 : Simplifying Rational Expressions

Example Question #1 : Simplifying Rational Expressions

Example Question #3 : Simplifying Rational Expressions

Example Question #4 : Simplifying Rational Expressions

Example Question #5 : Simplifying Rational Expressions

Example Question #1 : Solving Rational Equations And Inequalities

All High School Math Resources

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