Algebra Examples

$\frac{1}{3} x - 3 < - 1 \frac{5}{6}$

Step 1

Simplify the left side.

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Step 1.1

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

$\frac{x}{3} - 3 < - 1 \frac{5}{6}$

Step 2

Simplify the right side.

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Step 2.1

Convert $1 \frac{5}{6}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{x}{3} - 3 < - (1 + \frac{5}{6})$

Step 2.1.2

Add $1$ and $\frac{5}{6}$ .

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Step 2.1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{x}{3} - 3 < - (\frac{6}{6} + \frac{5}{6})$

Step 2.1.2.2

Combine the numerators over the common denominator.

$\frac{x}{3} - 3 < - \frac{6 + 5}{6}$

Step 2.1.2.3

Add $6$ and $5$ .

$\frac{x}{3} - 3 < - \frac{11}{6}$

Step 3

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

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Step 3.1

Add $3$ to both sides of the inequality.

$\frac{x}{3} < - \frac{11}{6} + 3$

Step 3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{3} < - \frac{11}{6} + 3 \cdot \frac{6}{6}$

Step 3.3

Combine $3$ and $\frac{6}{6}$ .

$\frac{x}{3} < - \frac{11}{6} + \frac{3 \cdot 6}{6}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x}{3} < \frac{- 11 + 3 \cdot 6}{6}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $6$ .

$\frac{x}{3} < \frac{- 11 + 18}{6}$

Step 3.5.2

Add $- 11$ and $18$ .

$\frac{x}{3} < \frac{7}{6}$

Step 4

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 < \frac{7}{6} \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 < \frac{7}{6} \cdot 3$

Step 5.1.1.2

Rewrite the expression.

$x < \frac{7}{6} \cdot 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Factor $3$ out of $6$ .

$x < \frac{7}{3 (2)} \cdot 3$

Step 5.2.1.2

Cancel the common factor.

$x < \frac{7}{3 \cdot 2} \cdot 3$

Step 5.2.1.3

Rewrite the expression.

$x < \frac{7}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{7}{2}$

Interval Notation:

$(- \infty, \frac{7}{2})$