Algebra Examples

$- 2 < \frac{x}{3} + 1 < 5$

Step 1

Solve $- 2 < \frac{x}{3} + 1$ for $x$ .

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Step 1.1

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

$\frac{x}{3} + 1 > - 2$

Step 1.2

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

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Step 1.2.1

Subtract $1$ from both sides of the inequality.

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

Step 1.2.2

Subtract $1$ from $- 2$ .

$\frac{x}{3} > - 3$

Step 1.3

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 > - 3 \cdot 3$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Cancel the common factor of $3$ .

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Step 1.4.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 > - 3 \cdot 3$

Step 1.4.1.1.2

Rewrite the expression.

$x > - 3 \cdot 3$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $- 3$ by $3$ .

$x > - 9$

Step 2

Solve $\frac{x}{3} + 1 < 5$ for $x$ .

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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.

$\frac{x}{3} < 5 - 1$

Step 2.1.2

Subtract $1$ from $5$ .

$\frac{x}{3} < 4$

Step 2.2

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 < 4 \cdot 3$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $3$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 < 4 \cdot 3$

Step 2.3.1.1.2

Rewrite the expression.

$x < 4 \cdot 3$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $4$ by $3$ .

$x < 12$

Step 3

Find the intersection of $x > - 9$ and $x < 12$ .

$- 9 < x < 12$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 12$

Interval Notation:

$(- 9, 12)$

Step 5