Algebra Examples

$\frac{2}{3} x - \frac{5}{6} < \frac{1}{2}$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} - \frac{5}{6} < \frac{1}{2}$

Step 2

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

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Step 2.1

Add $\frac{5}{6}$ to both sides of the inequality.

$\frac{2 x}{3} < \frac{1}{2} + \frac{5}{6}$

Step 2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x}{3} < \frac{1}{2} \cdot \frac{3}{3} + \frac{5}{6}$

Step 2.3

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

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Step 2.3.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{2 x}{3} < \frac{3}{2 \cdot 3} + \frac{5}{6}$

Step 2.3.2

Multiply $2$ by $3$ .

$\frac{2 x}{3} < \frac{3}{6} + \frac{5}{6}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2 x}{3} < \frac{3 + 5}{6}$

Step 2.5

Add $3$ and $5$ .

$\frac{2 x}{3} < \frac{8}{6}$

Step 2.6

Cancel the common factor of $8$ and $6$ .

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Step 2.6.1

Factor $2$ out of $8$ .

$\frac{2 x}{3} < \frac{2 (4)}{6}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $6$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x < 4$

Step 4

Divide each term in $2 x < 4$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x < 4$ by $2$ .

$\frac{2 x}{2} < \frac{4}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{4}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x < \frac{4}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $4$ by $2$ .

$x < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 2$

Interval Notation:

$(- \infty, 2)$