Algebra Examples

$\frac{3 x - 5}{4} - \frac{x - 6}{12} < 1$

Step 1

Simplify $\frac{3 x - 5}{4} - \frac{x - 6}{12}$ .

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Step 1.1

To write $\frac{3 x - 5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x - 5}{4} \cdot \frac{3}{3} - \frac{x - 6}{12} < 1$

Step 1.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{3 x - 5}{4}$ by $\frac{3}{3}$ .

$\frac{(3 x - 5) \cdot 3}{4 \cdot 3} - \frac{x - 6}{12} < 1$

Step 1.2.2

Multiply $4$ by $3$ .

$\frac{(3 x - 5) \cdot 3}{12} - \frac{x - 6}{12} < 1$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(3 x - 5) \cdot 3 - (x - 6)}{12} < 1$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{3 x \cdot 3 - 5 \cdot 3 - (x - 6)}{12} < 1$

Step 1.4.2

Multiply $3$ by $3$ .

$\frac{9 x - 5 \cdot 3 - (x - 6)}{12} < 1$

Step 1.4.3

Multiply $- 5$ by $3$ .

$\frac{9 x - 15 - (x - 6)}{12} < 1$

Step 1.4.4

Apply the distributive property.

$\frac{9 x - 15 - x - - 6}{12} < 1$

Step 1.4.5

Multiply $- 1$ by $- 6$ .

$\frac{9 x - 15 - x + 6}{12} < 1$

Step 1.4.6

Subtract $x$ from $9 x$ .

$\frac{8 x - 15 + 6}{12} < 1$

Step 1.4.7

Add $- 15$ and $6$ .

$\frac{8 x - 9}{12} < 1$

Step 2

Multiply both sides by $12$ .

$\frac{8 x - 9}{12} \cdot 12 < 1 \cdot 12$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{8 x - 9}{12} \cdot 12 < 1 \cdot 12$

Step 3.1.1.2

Rewrite the expression.

$8 x - 9 < 1 \cdot 12$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $12$ by $1$ .

$8 x - 9 < 12$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $9$ to both sides of the inequality.

$8 x < 12 + 9$

Step 4.1.2

Add $12$ and $9$ .

$8 x < 21$

Step 4.2

Divide each term in $8 x < 21$ by $8$ and simplify.

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Step 4.2.1

Divide each term in $8 x < 21$ by $8$ .

$\frac{8 x}{8} < \frac{21}{8}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} < \frac{21}{8}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{21}{8}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{21}{8}$

Interval Notation:

$(- \infty, \frac{21}{8})$