Finite Math Examples

$\frac{5 x + 3}{- 4} < - \frac{23}{5}$

Step 1

Multiply both sides by $- 4$ .

$\frac{5 x + 3}{- 4} \cdot - 4 > - \frac{23}{5} \cdot - 4$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{5 x + 3}{- 4} \cdot - 4$ .

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Step 2.1.1.1

Cancel the common factor of $4$ .

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Step 2.1.1.1.1

Factor $4$ out of $- 4$ .

$\frac{5 x + 3}{4 (- 1)} \cdot - 4 > - \frac{23}{5} \cdot - 4$

Step 2.1.1.1.2

Factor $4$ out of $- 4$ .

$\frac{5 x + 3}{4 \cdot - 1} \cdot (4 \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.1.3

Cancel the common factor.

$\frac{5 x + 3}{4 \cdot - 1} \cdot (4 \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.1.4

Rewrite the expression.

$\frac{5 x + 3}{- 1} \cdot - 1 > - \frac{23}{5} \cdot - 4$

Step 2.1.1.2

Combine $\frac{5 x + 3}{- 1}$ and $- 1$ .

$\frac{(5 x + 3) \cdot - 1}{- 1} > - \frac{23}{5} \cdot - 4$

Step 2.1.1.3

Simplify the expression.

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Step 2.1.1.3.1

Move the negative one from the denominator of $\frac{(5 x + 3) \cdot - 1}{- 1}$ .

$- 1 \cdot ((5 x + 3) \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.3.2

Rewrite $- 1 \cdot ((5 x + 3) \cdot - 1)$ as $- ((5 x + 3) \cdot - 1)$ .

$- ((5 x + 3) \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.4

Apply the distributive property.

$- (5 x \cdot - 1 + 3 \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.5

Multiply.

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Step 2.1.1.5.1

Multiply $- 1$ by $5$ .

$- (- 5 x + 3 \cdot - 1) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.5.2

Multiply $3$ by $- 1$ .

$- (- 5 x - 3) > - \frac{23}{5} \cdot - 4$

Step 2.1.1.6

Apply the distributive property.

$- (- 5 x) - - 3 > - \frac{23}{5} \cdot - 4$

Step 2.1.1.7

Multiply.

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Step 2.1.1.7.1

Multiply $- 5$ by $- 1$ .

$5 x - - 3 > - \frac{23}{5} \cdot - 4$

Step 2.1.1.7.2

Multiply $- 1$ by $- 3$ .

$5 x + 3 > - \frac{23}{5} \cdot - 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- \frac{23}{5} \cdot - 4$ .

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Step 2.2.1.1

Multiply $- 4$ by $- 1$ .

$5 x + 3 > 4 (\frac{23}{5})$

Step 2.2.1.2

Combine $4$ and $\frac{23}{5}$ .

$5 x + 3 > \frac{4 \cdot 23}{5}$

Step 2.2.1.3

Multiply $4$ by $23$ .

$5 x + 3 > \frac{92}{5}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3$ from both sides of the inequality.

$5 x > \frac{92}{5} - 3$

Step 3.1.2

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

$5 x > \frac{92}{5} - 3 \cdot \frac{5}{5}$

Step 3.1.3

Combine $- 3$ and $\frac{5}{5}$ .

$5 x > \frac{92}{5} + \frac{- 3 \cdot 5}{5}$

Step 3.1.4

Combine the numerators over the common denominator.

$5 x > \frac{92 - 3 \cdot 5}{5}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 3$ by $5$ .

$5 x > \frac{92 - 15}{5}$

Step 3.1.5.2

Subtract $15$ from $92$ .

$5 x > \frac{77}{5}$

Step 3.2

Divide each term in $5 x > \frac{77}{5}$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 x > \frac{77}{5}$ by $5$ .

$\frac{5 x}{5} > \frac{\frac{77}{5}}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{\frac{77}{5}}{5}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{\frac{77}{5}}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x > \frac{77}{5} \cdot \frac{1}{5}$

Step 3.2.3.2

Multiply $\frac{77}{5} \cdot \frac{1}{5}$ .

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Step 3.2.3.2.1

Multiply $\frac{77}{5}$ by $\frac{1}{5}$ .

$x > \frac{77}{5 \cdot 5}$

Step 3.2.3.2.2

Multiply $5$ by $5$ .

$x > \frac{77}{25}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{77}{25}$

Interval Notation:

$(\frac{77}{25}, \infty)$