Precalculus Examples

$\frac{1}{3} x - 2 < \frac{1}{4} x + 5$

Step 1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} - 2 < \frac{1}{4} x + 5$

Step 2

Combine $\frac{1}{4}$ and $x$ .

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

Step 3

Move all terms containing $x$ to the left side of the inequality.

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Step 3.1

Subtract $\frac{x}{4}$ from both sides of the inequality.

$\frac{x}{3} - 2 - \frac{x}{4} < 5$

Step 3.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} - 2 < 5$

Step 3.3

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} \cdot \frac{3}{3} - 2 < 5$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} \cdot \frac{3}{3} - 2 < 5$

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} - \frac{x}{4} \cdot \frac{3}{3} - 2 < 5$

Step 3.4.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3} - 2 < 5$

Step 3.4.4

Multiply $4$ by $3$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{12} - 2 < 5$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x \cdot 3}{12} - 2 < 5$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 3.6.1.1.1

Factor $x$ out of $x \cdot 4$ .

$\frac{x (4) - x \cdot 3}{12} - 2 < 5$

Step 3.6.1.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x (4) + x (- 1 \cdot 3)}{12} - 2 < 5$

Step 3.6.1.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

$\frac{x (4 - 1 \cdot 3)}{12} - 2 < 5$

Step 3.6.1.2

Multiply $- 1$ by $3$ .

$\frac{x (4 - 3)}{12} - 2 < 5$

Step 3.6.1.3

Subtract $3$ from $4$ .

$\frac{x \cdot 1}{12} - 2 < 5$

Step 3.6.2

Multiply $x$ by $1$ .

$\frac{x}{12} - 2 < 5$

Step 4

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.1

Add $2$ to both sides of the inequality.

$\frac{x}{12} < 5 + 2$

Step 4.2

Add $5$ and $2$ .

$\frac{x}{12} < 7$

Step 5

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 < 7 \cdot 12$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $12$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 < 7 \cdot 12$

Step 6.1.1.2

Rewrite the expression.

$x < 7 \cdot 12$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $7$ by $12$ .

$x < 84$

Step 7

Convert the inequality to interval notation.

$(- \infty, 84)$

Step 8