Algebra Examples

$\frac{3}{2} x - (- 6 x - 2) < 5$

Step 1

Simplify $\frac{3}{2} x - (- 6 x - 2)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} - (- 6 x - 2) < 5$

Step 1.1.2

Apply the distributive property.

$\frac{3 x}{2} - (- 6 x) - - 2 < 5$

Step 1.1.3

Multiply $- 6$ by $- 1$ .

$\frac{3 x}{2} + 6 x - - 2 < 5$

Step 1.1.4

Multiply $- 1$ by $- 2$ .

$\frac{3 x}{2} + 6 x + 2 < 5$

$\frac{3 x}{2} + 6 x + 2 < 5$

Step 1.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{2} + 6 x \cdot \frac{2}{2} + 2 < 5$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $6 x$ and $\frac{2}{2}$ .

$\frac{3 x}{2} + \frac{6 x \cdot 2}{2} + 2 < 5$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{3 x + 6 x \cdot 2}{2} + 2 < 5$

$\frac{3 x + 6 x \cdot 2}{2} + 2 < 5$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $3 x$ out of $3 x + 6 x \cdot 2$ .

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Step 1.4.1.1

Factor $3 x$ out of $3 x$ .

$\frac{3 x (1) + 6 x \cdot 2}{2} + 2 < 5$

Step 1.4.1.2

Factor $3 x$ out of $6 x \cdot 2$ .

$\frac{3 x (1) + 3 x (2 \cdot 2)}{2} + 2 < 5$

Step 1.4.1.3

Factor $3 x$ out of $3 x (1) + 3 x (2 \cdot 2)$ .

$\frac{3 x (1 + 2 \cdot 2)}{2} + 2 < 5$

$\frac{3 x (1 + 2 \cdot 2)}{2} + 2 < 5$

Step 1.4.2

Multiply $2$ by $2$ .

$\frac{3 x (1 + 4)}{2} + 2 < 5$

Step 1.4.3

Add $1$ and $4$ .

$\frac{3 x \cdot 5}{2} + 2 < 5$

Step 1.4.4

Multiply $5$ by $3$ .

$\frac{15 x}{2} + 2 < 5$

$\frac{15 x}{2} + 2 < 5$

$\frac{15 x}{2} + 2 < 5$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the inequality.

$\frac{15 x}{2} < 5 - 2$

Step 2.2

Subtract $2$ from $5$ .

$\frac{15 x}{2} < 3$

$\frac{15 x}{2} < 3$

Step 3

Multiply both sides by $2$ .

$\frac{15 x}{2} \cdot 2 < 3 \cdot 2$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{15 x}{2} \cdot 2 < 3 \cdot 2$

Step 4.1.1.2

Rewrite the expression.

$15 x < 3 \cdot 2$

$15 x < 3 \cdot 2$

$15 x < 3 \cdot 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $3$ by $2$ .

$15 x < 6$

$15 x < 6$

$15 x < 6$

Step 5

Divide each term in $15 x < 6$ by $15$ and simplify.

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Step 5.1

Divide each term in $15 x < 6$ by $15$ .

$\frac{15 x}{15} < \frac{6}{15}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $15$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{15 x}{15} < \frac{6}{15}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x < \frac{6}{15}$

$x < \frac{6}{15}$

$x < \frac{6}{15}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $6$ and $15$ .

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Step 5.3.1.1

Factor $3$ out of $6$ .

$x < \frac{3 (2)}{15}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $15$ .

$x < \frac{3 \cdot 2}{3 \cdot 5}$

Step 5.3.1.2.2

Cancel the common factor.

$x < \frac{3 \cdot 2}{3 \cdot 5}$

Step 5.3.1.2.3

Rewrite the expression.

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

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

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

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

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

Step 6

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, \frac{2}{5})$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$