Precalculus Examples

$\frac{1}{3} x + 2 < \frac{1}{6} x - 1$

Step 1

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

$\frac{x}{3} + 2 < \frac{1}{6} x - 1$

Step 2

Combine $\frac{1}{6}$ and $x$ .

$\frac{x}{3} + 2 < \frac{x}{6} - 1$

Step 3

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

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Step 3.1

Subtract $\frac{x}{6}$ from both sides of the inequality.

$\frac{x}{3} + 2 - \frac{x}{6} < - 1$

Step 3.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{3} \cdot \frac{2}{2} - \frac{x}{6} + 2 < - 1$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{3 \cdot 2} - \frac{x}{6} + 2 < - 1$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{x \cdot 2}{6} - \frac{x}{6} + 2 < - 1$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{6} + 2 < - 1$

Step 3.5

Subtract $x$ from $x \cdot 2$ .

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Step 3.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{6} + 2 < - 1$

Step 3.5.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{6} + 2 < - 1$

Step 4

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

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Step 4.1

Subtract $2$ from both sides of the inequality.

$\frac{x}{6} < - 1 - 2$

Step 4.2

Subtract $2$ from $- 1$ .

$\frac{x}{6} < - 3$

Step 5

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 < - 3 \cdot 6$

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 $6$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 < - 3 \cdot 6$

Step 6.1.1.2

Rewrite the expression.

$x < - 3 \cdot 6$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 3$ by $6$ .

$x < - 18$

Step 7

Convert the inequality to interval notation.

$(- \infty, - 18)$

Step 8