Algebra Examples

$- 1 < - (- 9 x - 1) - \frac{5}{2} x$

Step 1

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

$- (- 9 x - 1) - \frac{5}{2} x > - 1$

Step 2

Simplify $- (- 9 x - 1) - \frac{5}{2} x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- (- 9 x) - - 1 - \frac{5}{2} x > - 1$

Step 2.1.2

Multiply $- 9$ by $- 1$ .

$9 x - - 1 - \frac{5}{2} x > - 1$

Step 2.1.3

Multiply $- 1$ by $- 1$ .

$9 x + 1 - \frac{5}{2} x > - 1$

Step 2.1.4

Combine $x$ and $\frac{5}{2}$ .

$9 x + 1 - \frac{x \cdot 5}{2} > - 1$

Step 2.1.5

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

$9 x + 1 - \frac{5 x}{2} > - 1$

Step 2.2

To write $9 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 x \cdot \frac{2}{2} - \frac{5 x}{2} + 1 > - 1$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $9 x$ and $\frac{2}{2}$ .

$\frac{9 x \cdot 2}{2} - \frac{5 x}{2} + 1 > - 1$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{9 x \cdot 2 - 5 x}{2} + 1 > - 1$

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 $9 x \cdot 2 - 5 x$ .

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Step 2.4.1.1.1

Factor $x$ out of $9 x \cdot 2$ .

$\frac{x (9 \cdot 2) - 5 x}{2} + 1 > - 1$

Step 2.4.1.1.2

Factor $x$ out of $- 5 x$ .

$\frac{x (9 \cdot 2) + x \cdot - 5}{2} + 1 > - 1$

Step 2.4.1.1.3

Factor $x$ out of $x (9 \cdot 2) + x \cdot - 5$ .

$\frac{x (9 \cdot 2 - 5)}{2} + 1 > - 1$

Step 2.4.1.2

Multiply $9$ by $2$ .

$\frac{x (18 - 5)}{2} + 1 > - 1$

Step 2.4.1.3

Subtract $5$ from $18$ .

$\frac{x \cdot 13}{2} + 1 > - 1$

Step 2.4.2

Move $13$ to the left of $x$ .

$\frac{13 x}{2} + 1 > - 1$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the inequality.

$\frac{13 x}{2} > - 1 - 1$

Step 3.2

Subtract $1$ from $- 1$ .

$\frac{13 x}{2} > - 2$

Step 4

Multiply both sides by $2$ .

$\frac{13 x}{2} \cdot 2 > - 2 \cdot 2$

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.

$\frac{13 x}{2} \cdot 2 > - 2 \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$13 x > - 2 \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 2$ by $2$ .

$13 x > - 4$

Step 6

Divide each term in $13 x > - 4$ by $13$ and simplify.

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Step 6.1

Divide each term in $13 x > - 4$ by $13$ .

$\frac{13 x}{13} > \frac{- 4}{13}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $13$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{13 x}{13} > \frac{- 4}{13}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 4}{13}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x > - \frac{4}{13}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{4}{13}$

Interval Notation:

$(- \frac{4}{13}, \infty)$