Algebra Examples

$- 5 x + 9 < - 6$ or $- 3 x + 1 \geq 7$

Step 1

Simplify the first inequality.

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Step 1.1

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

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Step 1.1.1

Subtract $9$ from both sides of the inequality.

$- 5 x < - 6 - 9$ or $- 3 x + 1 \geq 7$

Step 1.1.2

Subtract $9$ from $- 6$ .

$- 5 x < - 15$ or $- 3 x + 1 \geq 7$

Step 1.2

Divide each term in $- 5 x < - 15$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 x < - 15$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{- 15}{- 5}$ or $- 3 x + 1 \geq 7$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{- 15}{- 5}$ or $- 3 x + 1 \geq 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 15}{- 5}$ or $- 3 x + 1 \geq 7$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 15$ by $- 5$ .

$x > 3$ or $- 3 x + 1 \geq 7$

Step 2

Simplify the second inequality.

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Step 2.1

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

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Step 2.1.1

Subtract $1$ from both sides of the inequality.

$x > 3$ or $- 3 x \geq 7 - 1$

Step 2.1.2

Subtract $1$ from $7$ .

$x > 3$ or $- 3 x \geq 6$

Step 2.2

Divide each term in $- 3 x \geq 6$ by $- 3$ and simplify.

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Step 2.2.1

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

$x > 3$ or $\frac{- 3 x}{- 3} \leq \frac{6}{- 3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$x > 3$ or $\frac{- 3 x}{- 3} \leq \frac{6}{- 3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x > 3$ or $x \leq \frac{6}{- 3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $6$ by $- 3$ .

$x > 3$ or $x \leq - 2$

Step 3

The union consists of all of the elements that are contained in each interval.

$x \leq - 2$ or $x > 3$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2 or x > 3$

Interval Notation:

$(- \infty, - 2] \cup (3, \infty)$

Step 5