Algebra Examples

$2 x + 1 \leq - 6$ or $2 x + 1 \geq 6$

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 $1$ from both sides of the inequality.

$2 x \leq - 6 - 1$

Step 1.1.2

Subtract $1$ from $- 6$ .

$2 x \leq - 7$

Step 1.2

Divide each term in $2 x \leq - 7$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x \leq - 7$ by $2$ .

$\frac{2 x}{2} \leq \frac{- 7}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{- 7}{2}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{7}{2}$

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.

$2 x \geq 6 - 1$

Step 2.1.2

Subtract $1$ from $6$ .

$2 x \geq 5$

Step 2.2

Divide each term in $2 x \geq 5$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq 5$ by $2$ .

$\frac{2 x}{2} \geq \frac{5}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{5}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{5}{2}$

Step 3

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

$x \leq - \frac{7}{2}$ or $x \geq \frac{5}{2}$

Step 4

Convert the inequality to interval notation.

$(- \infty, - \frac{7}{2}] \cup [\frac{5}{2}, \infty)$

Step 5