Algebra Examples

$20 x - 18 < 50 < 36 x - 16$

Step 1

Solve $20 x - 18 < 50$ for $x$ .

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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

Add $18$ to both sides of the inequality.

$20 x < 50 + 18$

Step 1.1.2

Add $50$ and $18$ .

$20 x < 68$

Step 1.2

Divide each term in $20 x < 68$ by $20$ and simplify.

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Step 1.2.1

Divide each term in $20 x < 68$ by $20$ .

$\frac{20 x}{20} < \frac{68}{20}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $20$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} < \frac{68}{20}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{68}{20}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $68$ and $20$ .

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Step 1.2.3.1.1

Factor $4$ out of $68$ .

$x < \frac{4 (17)}{20}$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $4$ out of $20$ .

$x < \frac{4 \cdot 17}{4 \cdot 5}$

Step 1.2.3.1.2.2

Cancel the common factor.

$x < \frac{4 \cdot 17}{4 \cdot 5}$

Step 1.2.3.1.2.3

Rewrite the expression.

$x < \frac{17}{5}$

Step 2

Solve $50 < 36 x - 16$ for $x$ .

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Step 2.1

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

$36 x - 16 > 50$

Step 2.2

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

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Step 2.2.1

Add $16$ to both sides of the inequality.

$36 x > 50 + 16$

Step 2.2.2

Add $50$ and $16$ .

$36 x > 66$

Step 2.3

Divide each term in $36 x > 66$ by $36$ and simplify.

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Step 2.3.1

Divide each term in $36 x > 66$ by $36$ .

$\frac{36 x}{36} > \frac{66}{36}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $36$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{36 x}{36} > \frac{66}{36}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x > \frac{66}{36}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $66$ and $36$ .

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Step 2.3.3.1.1

Factor $6$ out of $66$ .

$x > \frac{6 (11)}{36}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $6$ out of $36$ .

$x > \frac{6 \cdot 11}{6 \cdot 6}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x > \frac{6 \cdot 11}{6 \cdot 6}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x > \frac{11}{6}$

Step 3

Find the intersection of $x < \frac{17}{5}$ and $x > \frac{11}{6}$ .

$\frac{11}{6} < x < \frac{17}{5}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$\frac{11}{6} < x < \frac{17}{5}$

Interval Notation:

$(\frac{11}{6}, \frac{17}{5})$

Step 5