Algebra Examples

$- \frac{1}{2} x \geq - 2 x - (- \frac{3}{4} x + 1)$

Step 1

Rewrite so $x$ is on the left side of the inequality.

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $x$ and $\frac{3}{4}$ .

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

Step 2.1.1.2

Move $3$ to the left of $x$ .

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

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

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Step 2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.3.2

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

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

Step 2.1.4

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $- 2 x$ and $\frac{4}{4}$ .

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

Step 2.3.2

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $x$ out of $- 2 x \cdot 4 + 3 x$ .

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Step 2.4.1.1.1

Factor $x$ out of $- 2 x \cdot 4$ .

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

Step 2.4.1.1.2

Factor $x$ out of $3 x$ .

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

Step 2.4.1.1.3

Factor $x$ out of $x (- 2 \cdot 4) + x \cdot 3$ .

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

Step 2.4.1.2

Multiply $- 2$ by $4$ .

$\frac{x (- 8 + 3)}{4} - 1 \leq - \frac{1}{2} x$

Step 2.4.1.3

Add $- 8$ and $3$ .

$\frac{x \cdot - 5}{4} - 1 \leq - \frac{1}{2} x$

Step 2.4.2

Move $- 5$ to the left of $x$ .

$\frac{- 5 \cdot x}{4} - 1 \leq - \frac{1}{2} x$

Step 2.4.3

Move the negative in front of the fraction.

$- \frac{5 x}{4} - 1 \leq - \frac{1}{2} x$

Step 3

Combine $x$ and $\frac{1}{2}$ .

$- \frac{5 x}{4} - 1 \leq - \frac{x}{2}$

Step 4

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

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Step 4.1

Add $\frac{x}{2}$ to both sides of the inequality.

$- \frac{5 x}{4} - 1 + \frac{x}{2} \leq 0$

Step 4.2

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

$- \frac{5 x}{4} + \frac{x}{2} \cdot \frac{2}{2} - 1 \leq 0$

Step 4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

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

$- \frac{5 x}{4} + \frac{x \cdot 2}{2 \cdot 2} - 1 \leq 0$

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{5 x}{4} + \frac{x \cdot 2}{4} - 1 \leq 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 5 x + x \cdot 2}{4} - 1 \leq 0$

Step 4.5

Add $- 5 x$ and $x \cdot 2$ .

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Step 4.5.1

Reorder $x$ and $2$ .

$\frac{- 5 x + 2 \cdot x}{4} - 1 \leq 0$

Step 4.5.2

Add $- 5 x$ and $2 \cdot x$ .

$\frac{- 3 x}{4} - 1 \leq 0$

Step 4.6

Move the negative in front of the fraction.

$- \frac{3 x}{4} - 1 \leq 0$

Step 5

Add $1$ to both sides of the inequality.

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

Step 6

Divide each term in $- \frac{3 x}{4} \leq 1$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- \frac{3 x}{4} \leq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 x}{4}}{- 1} \geq \frac{1}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{4}}{1} \geq \frac{1}{- 1}$

Step 6.2.2

Divide $\frac{3 x}{4}$ by $1$ .

$\frac{3 x}{4} \geq \frac{1}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $1$ by $- 1$ .

$\frac{3 x}{4} \geq - 1$

Step 7

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 \geq - 1 \cdot 4$

Step 8

Simplify.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Cancel the common factor of $4$ .

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Step 8.1.1.1

Cancel the common factor.

$\frac{3 x}{4} \cdot 4 \geq - 1 \cdot 4$

Step 8.1.1.2

Rewrite the expression.

$3 x \geq - 1 \cdot 4$

Step 8.2

Simplify the right side.

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Step 8.2.1

Multiply $- 1$ by $4$ .

$3 x \geq - 4$

Step 9

Divide each term in $3 x \geq - 4$ by $3$ and simplify.

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Step 9.1

Divide each term in $3 x \geq - 4$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $3$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 4}{3}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x \geq - \frac{4}{3}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{4}{3}$

Interval Notation:

$[- \frac{4}{3}, \infty)$