Algebra Examples

$x \geq - 6 - \frac{1}{4} (- 6 x + 5)$

Step 1

Simplify $- 6 - \frac{1}{4} \cdot (- 6 x + 5)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$x \geq - 6 - \frac{1}{4} (- 6 x) - \frac{1}{4} \cdot 5$

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$x \geq - 6 + \frac{- 1}{4} (- 6 x) - \frac{1}{4} \cdot 5$

Step 1.1.2.2

Factor $2$ out of $4$ .

$x \geq - 6 + \frac{- 1}{2 (2)} (- 6 x) - \frac{1}{4} \cdot 5$

Step 1.1.2.3

Factor $2$ out of $- 6 x$ .

$x \geq - 6 + \frac{- 1}{2 (2)} (2 (- 3 x)) - \frac{1}{4} \cdot 5$

Step 1.1.2.4

Cancel the common factor.

$x \geq - 6 + \frac{- 1}{2 \cdot 2} (2 (- 3 x)) - \frac{1}{4} \cdot 5$

Step 1.1.2.5

Rewrite the expression.

$x \geq - 6 + \frac{- 1}{2} (- 3 x) - \frac{1}{4} \cdot 5$

Step 1.1.3

Combine $- 3$ and $\frac{- 1}{2}$ .

$x \geq - 6 + \frac{- 3 \cdot - 1}{2} x - \frac{1}{4} \cdot 5$

Step 1.1.4

Multiply $- 3$ by $- 1$ .

$x \geq - 6 + \frac{3}{2} x - \frac{1}{4} \cdot 5$

Step 1.1.5

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

$x \geq - 6 + \frac{3 x}{2} - \frac{1}{4} \cdot 5$

Step 1.1.6

Multiply $- \frac{1}{4} \cdot 5$ .

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Step 1.1.6.1

Multiply $5$ by $- 1$ .

$x \geq - 6 + \frac{3 x}{2} - 5 (\frac{1}{4})$

Step 1.1.6.2

Combine $- 5$ and $\frac{1}{4}$ .

$x \geq - 6 + \frac{3 x}{2} + \frac{- 5}{4}$

Step 1.1.7

Move the negative in front of the fraction.

$x \geq - 6 + \frac{3 x}{2} - \frac{5}{4}$

Step 1.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x \geq \frac{3 x}{2} - 6 \cdot \frac{4}{4} - \frac{5}{4}$

Step 1.3

Combine $- 6$ and $\frac{4}{4}$ .

$x \geq \frac{3 x}{2} + \frac{- 6 \cdot 4}{4} - \frac{5}{4}$

Step 1.4

Combine the numerators over the common denominator.

$x \geq \frac{3 x}{2} + \frac{- 6 \cdot 4 - 5}{4}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 6$ by $4$ .

$x \geq \frac{3 x}{2} + \frac{- 24 - 5}{4}$

Step 1.5.2

Subtract $5$ from $- 24$ .

$x \geq \frac{3 x}{2} + \frac{- 29}{4}$

Step 1.6

Move the negative in front of the fraction.

$x \geq \frac{3 x}{2} - \frac{29}{4}$

Step 2

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

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Step 2.1

Subtract $\frac{3 x}{2}$ from both sides of the inequality.

$x - \frac{3 x}{2} \geq - \frac{29}{4}$

Step 2.2

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

$x \cdot \frac{2}{2} - \frac{3 x}{2} \geq - \frac{29}{4}$

Step 2.3

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

$\frac{x \cdot 2}{2} - \frac{3 x}{2} \geq - \frac{29}{4}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - 3 x}{2} \geq - \frac{29}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Move $2$ to the left of $x$ .

$\frac{2 \cdot x - 3 x}{2} \geq - \frac{29}{4}$

Step 2.5.2

Subtract $3 x$ from $2 x$ .

$\frac{- x}{2} \geq - \frac{29}{4}$

Step 2.6

Move the negative in front of the fraction.

$- \frac{x}{2} \geq - \frac{29}{4}$

Step 3

Divide each term in $- \frac{x}{2} \geq - \frac{29}{4}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{x}{2} \geq - \frac{29}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{2}}{- 1} \leq \frac{- \frac{29}{4}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} \leq \frac{- \frac{29}{4}}{- 1}$

Step 3.2.2

Divide $\frac{x}{2}$ by $1$ .

$\frac{x}{2} \leq \frac{- \frac{29}{4}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\frac{x}{2} \leq \frac{\frac{29}{4}}{1}$

Step 3.3.2

Divide $\frac{29}{4}$ by $1$ .

$\frac{x}{2} \leq \frac{29}{4}$

Step 4

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \leq \frac{29}{4} \cdot 2$

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 $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \leq \frac{29}{4} \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$x \leq \frac{29}{4} \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Factor $2$ out of $4$ .

$x \leq \frac{29}{2 (2)} \cdot 2$

Step 5.2.1.2

Cancel the common factor.

$x \leq \frac{29}{2 \cdot 2} \cdot 2$

Step 5.2.1.3

Rewrite the expression.

$x \leq \frac{29}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{29}{2}$

Interval Notation:

$(- \infty, \frac{29}{2}]$