Algebra Examples

$\frac{3 + x}{4} + \frac{2 - x}{3} < 0$

Step 1

Simplify $\frac{3 + x}{4} + \frac{2 - x}{3}$ .

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Step 1.1

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

$\frac{3 + x}{4} \cdot \frac{3}{3} + \frac{2 - x}{3} < 0$

Step 1.2

To write $\frac{2 - x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 + x}{4} \cdot \frac{3}{3} + \frac{2 - x}{3} \cdot \frac{4}{4} < 0$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{3 + x}{4}$ by $\frac{3}{3}$ .

$\frac{(3 + x) \cdot 3}{4 \cdot 3} + \frac{2 - x}{3} \cdot \frac{4}{4} < 0$

Step 1.3.2

Multiply $4$ by $3$ .

$\frac{(3 + x) \cdot 3}{12} + \frac{2 - x}{3} \cdot \frac{4}{4} < 0$

Step 1.3.3

Multiply $\frac{2 - x}{3}$ by $\frac{4}{4}$ .

$\frac{(3 + x) \cdot 3}{12} + \frac{(2 - x) \cdot 4}{3 \cdot 4} < 0$

Step 1.3.4

Multiply $3$ by $4$ .

$\frac{(3 + x) \cdot 3}{12} + \frac{(2 - x) \cdot 4}{12} < 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(3 + x) \cdot 3 + (2 - x) \cdot 4}{12} < 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{3 \cdot 3 + x \cdot 3 + (2 - x) \cdot 4}{12} < 0$

Step 1.5.2

Multiply $3$ by $3$ .

$\frac{9 + x \cdot 3 + (2 - x) \cdot 4}{12} < 0$

Step 1.5.3

Move $3$ to the left of $x$ .

$\frac{9 + 3 \cdot x + (2 - x) \cdot 4}{12} < 0$

Step 1.5.4

Apply the distributive property.

$\frac{9 + 3 x + 2 \cdot 4 - x \cdot 4}{12} < 0$

Step 1.5.5

Multiply $2$ by $4$ .

$\frac{9 + 3 x + 8 - x \cdot 4}{12} < 0$

Step 1.5.6

Multiply $4$ by $- 1$ .

$\frac{9 + 3 x + 8 - 4 x}{12} < 0$

Step 1.5.7

Add $9$ and $8$ .

$\frac{3 x + 17 - 4 x}{12} < 0$

Step 1.5.8

Subtract $4 x$ from $3 x$ .

$\frac{- x + 17}{12} < 0$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 17}{12} < 0$

Step 1.6.2

Rewrite $17$ as $- 1 (- 17)$ .

$\frac{- (x) - 1 (- 17)}{12} < 0$

Step 1.6.3

Factor $- 1$ out of $- (x) - 1 (- 17)$ .

$\frac{- (x - 17)}{12} < 0$

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (x - 17)$ as $- 1 (x - 17)$ .

$\frac{- 1 (x - 17)}{12} < 0$

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{x - 17}{12} < 0$

Step 2

Divide each term in $- \frac{x - 17}{12} < 0$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{x - 17}{12} < 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 17}{12}}{- 1} > \frac{0}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 17}{12}}{1} > \frac{0}{- 1}$

Step 2.2.2

Divide $\frac{x - 17}{12}$ by $1$ .

$\frac{x - 17}{12} > \frac{0}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 1$ .

$\frac{x - 17}{12} > 0$

Step 3

Multiply both sides by $12$ .

$\frac{x - 17}{12} \cdot 12 > 0 \cdot 12$

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 $12$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{x - 17}{12} \cdot 12 > 0 \cdot 12$

Step 4.1.1.2

Rewrite the expression.

$x - 17 > 0 \cdot 12$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $0$ by $12$ .

$x - 17 > 0$

Step 5

Add $17$ to both sides of the inequality.

$x > 17$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 17$

Interval Notation:

$(17, \infty)$