Algebra Examples

$- x + 3 \geq 1$ or $6 x + 4 < - 3$

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 $3$ from both sides of the inequality.

$- x \geq 1 - 3$ or $6 x + 4 < - 3$

Step 1.1.2

Subtract $3$ from $1$ .

$- x \geq - 2$ or $6 x + 4 < - 3$

Step 1.2

Divide each term in $- x \geq - 2$ by $- 1$ and simplify.

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Step 1.2.1

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

$\frac{- x}{- 1} \leq \frac{- 2}{- 1}$ or $6 x + 4 < - 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 2}{- 1}$ or $6 x + 4 < - 3$

Step 1.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 1}$ or $6 x + 4 < - 3$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 2$ by $- 1$ .

$x \leq 2$ or $6 x + 4 < - 3$

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 $4$ from both sides of the inequality.

$x \leq 2$ or $6 x < - 3 - 4$

Step 2.1.2

Subtract $4$ from $- 3$ .

$x \leq 2$ or $6 x < - 7$

Step 2.2

Divide each term in $6 x < - 7$ by $6$ and simplify.

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Step 2.2.1

Divide each term in $6 x < - 7$ by $6$ .

$x \leq 2$ or $\frac{6 x}{6} < \frac{- 7}{6}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $6$ .

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Step 2.2.2.1.1

Cancel the common factor.

$x \leq 2$ or $\frac{6 x}{6} < \frac{- 7}{6}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq 2$ or $x < \frac{- 7}{6}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \leq 2$ or $x < - \frac{7}{6}$

Step 3

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

$x \leq 2$

Step 4

Convert the inequality to interval notation.

$(- \infty, 2]$

Step 5