Basic Math Examples

∣ ∣ ∣ \frac{3}{4} a - 7 ∣ ∣ ∣ = 5

$| \frac{3}{4} a - 7 | = 5$

Step 1

Remove the absolute value term. This creates a

\pm

$\pm$ on the right side of the equation because

| x | = \pm x

$| x | = \pm x$ .

\frac{3}{4} a - 7 = \pm 5

$\frac{3}{4} a - 7 = \pm 5$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

\frac{3}{4} a - 7 = 5

$\frac{3}{4} a - 7 = 5$

Step 2.2

Combine

\frac{3}{4}

$\frac{3}{4}$ and

a

$a$ .

\frac{3 a}{4} - 7 = 5

$\frac{3 a}{4} - 7 = 5$

Step 2.3

Move all terms not containing

a

$a$ to the right side of the equation.

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Step 2.3.1

Add

7

$7$ to both sides of the equation.

\frac{3 a}{4} = 5 + 7

$\frac{3 a}{4} = 5 + 7$

Step 2.3.2

Add

5

$5$ and

7

$7$ .

\frac{3 a}{4} = 12

$\frac{3 a}{4} = 12$

\frac{3 a}{4} = 12

$\frac{3 a}{4} = 12$

Step 2.4

Multiply both sides of the equation by

\frac{4}{3}

$\frac{4}{3}$ .

\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} \cdot 12

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} \cdot 12$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify

\frac{4}{3} \cdot \frac{3 a}{4}

$\frac{4}{3} \cdot \frac{3 a}{4}$ .

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Step 2.5.1.1.1

Cancel the common factor of

4

$4$ .

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Step 2.5.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} \cdot 12$

Step 2.5.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 a) = \frac{4}{3} \cdot 12$

Step 2.5.1.1.2

Cancel the common factor of

$3$ .

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Step 2.5.1.1.2.1

Factor

$3$ out of

$3 a$ .

$\frac{1}{3} (3 (a)) = \frac{4}{3} \cdot 12$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 a) = \frac{4}{3} \cdot 12$

Step 2.5.1.1.2.3

Rewrite the expression.

$a = \frac{4}{3} \cdot 12$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify

$\frac{4}{3} \cdot 12$ .

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Step 2.5.2.1.1

Cancel the common factor of

$3$ .

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Step 2.5.2.1.1.1

Factor

$3$ out of

$12$ .

$a = \frac{4}{3} \cdot (3 (4))$

Step 2.5.2.1.1.2

Cancel the common factor.

$a = \frac{4}{3} \cdot (3 \cdot 4)$

Step 2.5.2.1.1.3

Rewrite the expression.

$a = 4 \cdot 4$

Step 2.5.2.1.2

Multiply

$4$ by

$4$ .

$a = 16$

Step 2.6

Next, use the negative value of the

$\pm$ to find the second solution.

$\frac{3}{4} a - 7 = - 5$

Step 2.7

Combine

$\frac{3}{4}$ and

$a$ .

$\frac{3 a}{4} - 7 = - 5$

Step 2.8

Move all terms not containing

$a$ to the right side of the equation.

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Step 2.8.1

Add

$7$ to both sides of the equation.

$\frac{3 a}{4} = - 5 + 7$

Step 2.8.2

Add

$- 5$ and

$7$ .

$\frac{3 a}{4} = 2$

Step 2.9

Multiply both sides of the equation by

$\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} \cdot 2$

Step 2.10

Simplify both sides of the equation.

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Step 2.10.1

Simplify the left side.

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Step 2.10.1.1

Simplify

$\frac{4}{3} \cdot \frac{3 a}{4}$ .

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Step 2.10.1.1.1

Cancel the common factor of

$4$ .

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Step 2.10.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 a}{4} = \frac{4}{3} \cdot 2$

Step 2.10.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 a) = \frac{4}{3} \cdot 2$

Step 2.10.1.1.2

Cancel the common factor of

$3$ .

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Step 2.10.1.1.2.1

Factor

$3$ out of

$3 a$ .

$\frac{1}{3} (3 (a)) = \frac{4}{3} \cdot 2$

Step 2.10.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 a) = \frac{4}{3} \cdot 2$

Step 2.10.1.1.2.3

Rewrite the expression.

$a = \frac{4}{3} \cdot 2$

Step 2.10.2

Simplify the right side.

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Step 2.10.2.1

Multiply

$\frac{4}{3} \cdot 2$ .

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Step 2.10.2.1.1

Combine

$\frac{4}{3}$ and

$2$ .

$a = \frac{4 \cdot 2}{3}$

Step 2.10.2.1.2

Multiply

$4$ by

$2$ .

$a = \frac{8}{3}$

Step 2.11

The complete solution is the result of both the positive and negative portions of the solution.

$a = 16, \frac{8}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$a = 16, \frac{8}{3}$

Decimal Form:

$a = 16, 2.\overline{6}$

Mixed Number Form:

$a = 16, 2 \frac{2}{3}$