Algebra Examples

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

Step 1

Simplify $\frac{2 x}{3} - \frac{x + 1}{4}$ .

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Step 1.1

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 x}{3} \cdot \frac{4}{4} - \frac{x + 1}{4} \leq - 2$

Step 1.2

To write $- \frac{x + 1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x}{3} \cdot \frac{4}{4} - \frac{x + 1}{4} \cdot \frac{3}{3} \leq - 2$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{2 x}{3}$ by $\frac{4}{4}$ .

$\frac{2 x \cdot 4}{3 \cdot 4} - \frac{x + 1}{4} \cdot \frac{3}{3} \leq - 2$

Step 1.3.2

Multiply $3$ by $4$ .

$\frac{2 x \cdot 4}{12} - \frac{x + 1}{4} \cdot \frac{3}{3} \leq - 2$

Step 1.3.3

Multiply $\frac{x + 1}{4}$ by $\frac{3}{3}$ .

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

Step 1.3.4

Multiply $4$ by $3$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $4$ by $2$ .

$\frac{8 x - (x + 1) \cdot 3}{12} \leq - 2$

Step 1.5.2

Apply the distributive property.

$\frac{8 x + (- x - 1 \cdot 1) \cdot 3}{12} \leq - 2$

Step 1.5.3

Multiply $- 1$ by $1$ .

$\frac{8 x + (- x - 1) \cdot 3}{12} \leq - 2$

Step 1.5.4

Apply the distributive property.

$\frac{8 x - x \cdot 3 - 1 \cdot 3}{12} \leq - 2$

Step 1.5.5

Multiply $3$ by $- 1$ .

$\frac{8 x - 3 x - 1 \cdot 3}{12} \leq - 2$

Step 1.5.6

Multiply $- 1$ by $3$ .

$\frac{8 x - 3 x - 3}{12} \leq - 2$

Step 1.5.7

Subtract $3 x$ from $8 x$ .

$\frac{5 x - 3}{12} \leq - 2$

Step 2

Multiply both sides by $12$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$5 x - 3 \leq - 2 \cdot 12$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $- 2$ by $12$ .

$5 x - 3 \leq - 24$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $3$ to both sides of the inequality.

$5 x \leq - 24 + 3$

Step 4.1.2

Add $- 24$ and $3$ .

$5 x \leq - 21$

Step 4.2

Divide each term in $5 x \leq - 21$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x \leq - 21$ by $5$ .

$\frac{5 x}{5} \leq \frac{- 21}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{- 21}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 21}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{21}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{21}{5}$

Interval Notation:

$(- \infty, - \frac{21}{5}]$