Algebra Examples

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

Step 1

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

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

Step 2

Simplify $- (\frac{3}{2} x + 6) + x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Multiply $- 1$ by $6$ .

$- \frac{3 x}{2} - 6 + x > \frac{1}{4} x$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $- 3 x$ and $x \cdot 2$ .

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Step 2.5.1

Reorder $x$ and $2$ .

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

Step 2.5.2

Add $- 3 x$ and $2 \cdot x$ .

$\frac{- x}{2} - 6 > \frac{1}{4} x$

Step 2.6

Move the negative in front of the fraction.

$- \frac{x}{2} - 6 > \frac{1}{4} x$

Step 3

Combine $\frac{1}{4}$ and $x$ .

$- \frac{x}{2} - 6 > \frac{x}{4}$

Step 4

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

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Step 4.1

Subtract $\frac{x}{4}$ from both sides of the inequality.

$- \frac{x}{2} - 6 - \frac{x}{4} > 0$

Step 4.2

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

$- \frac{x}{2} \cdot \frac{2}{2} - \frac{x}{4} - 6 > 0$

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 $\frac{2}{2}$ .

$- \frac{x \cdot 2}{2 \cdot 2} - \frac{x}{4} - 6 > 0$

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{x \cdot 2}{4} - \frac{x}{4} - 6 > 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- x \cdot 2 - x}{4} - 6 > 0$

Step 4.5

Simplify each term.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

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

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Step 4.5.1.1.1

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

$\frac{x (- 1 \cdot 2) - x}{4} - 6 > 0$

Step 4.5.1.1.2

Factor $x$ out of $- x$ .

$\frac{x (- 1 \cdot 2) + x \cdot - 1}{4} - 6 > 0$

Step 4.5.1.1.3

Factor $x$ out of $x (- 1 \cdot 2) + x \cdot - 1$ .

$\frac{x (- 1 \cdot 2 - 1)}{4} - 6 > 0$

Step 4.5.1.2

Multiply $- 1$ by $2$ .

$\frac{x (- 2 - 1)}{4} - 6 > 0$

Step 4.5.1.3

Subtract $1$ from $- 2$ .

$\frac{x \cdot - 3}{4} - 6 > 0$

Step 4.5.2

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

$\frac{- 3 \cdot x}{4} - 6 > 0$

Step 4.5.3

Move the negative in front of the fraction.

$- \frac{3 x}{4} - 6 > 0$

Step 5

Add $6$ to both sides of the inequality.

$- \frac{3 x}{4} > 6$

Step 6

Divide each term in $- \frac{3 x}{4} > 6$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- \frac{3 x}{4} > 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 x}{4}}{- 1} < \frac{6}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{4}}{1} < \frac{6}{- 1}$

Step 6.2.2

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

$\frac{3 x}{4} < \frac{6}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $6$ by $- 1$ .

$\frac{3 x}{4} < - 6$

Step 7

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 < - 6 \cdot 4$

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.

$\frac{3 x}{4} \cdot 4 < - 6 \cdot 4$

Step 8.1.1.2

Rewrite the expression.

$3 x < - 6 \cdot 4$

Step 8.2

Simplify the right side.

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Step 8.2.1

Multiply $- 6$ by $4$ .

$3 x < - 24$

Step 9

Divide each term in $3 x < - 24$ by $3$ and simplify.

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Step 9.1

Divide each term in $3 x < - 24$ by $3$ .

$\frac{3 x}{3} < \frac{- 24}{3}$

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.

$\frac{3 x}{3} < \frac{- 24}{3}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 24}{3}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $- 24$ by $3$ .

$x < - 8$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < - 8$

Interval Notation:

$(- \infty, - 8)$