Algebra Examples

$11 x + 4 < 15$ or $12 x - 7 > - 25$

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 $4$ from both sides of the inequality.

$11 x < 15 - 4$ or $12 x - 7 > - 25$

Step 1.1.2

Subtract $4$ from $15$ .

$11 x < 11$ or $12 x - 7 > - 25$

Step 1.2

Divide each term in $11 x < 11$ by $11$ and simplify.

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Step 1.2.1

Divide each term in $11 x < 11$ by $11$ .

$\frac{11 x}{11} < \frac{11}{11}$ or $12 x - 7 > - 25$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $11$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} < \frac{11}{11}$ or $12 x - 7 > - 25$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{11}{11}$ or $12 x - 7 > - 25$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $11$ by $11$ .

$x < 1$ or $12 x - 7 > - 25$

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

Add $7$ to both sides of the inequality.

$x < 1$ or $12 x > - 25 + 7$

Step 2.1.2

Add $- 25$ and $7$ .

$x < 1$ or $12 x > - 18$

Step 2.2

Divide each term in $12 x > - 18$ by $12$ and simplify.

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Step 2.2.1

Divide each term in $12 x > - 18$ by $12$ .

$x < 1$ or $\frac{12 x}{12} > \frac{- 18}{12}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $12$ .

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Step 2.2.2.1.1

Cancel the common factor.

$x < 1$ or $\frac{12 x}{12} > \frac{- 18}{12}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < 1$ or $x > \frac{- 18}{12}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 18$ and $12$ .

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Step 2.2.3.1.1

Factor $6$ out of $- 18$ .

$x < 1$ or $x > \frac{6 (- 3)}{12}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $6$ out of $12$ .

$x < 1$ or $x > \frac{6 \cdot - 3}{6 \cdot 2}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x < 1$ or $x > \frac{6 \cdot - 3}{6 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x < 1$ or $x > \frac{- 3}{2}$

Step 2.2.3.2

Move the negative in front of the fraction.

$x < 1$ or $x > - \frac{3}{2}$

Step 3

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 5