Solve for x |5x-3|<12

$| 5 x - 3 | < 12$

Step 1

Write $| 5 x - 3 | < 12$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$5 x - 3 \geq 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$5 x \geq 3$

Divide each term in $5 x \geq 3$ by $5$ and simplify.

Divide each term in $5 x \geq 3$ by $5$ .

$\frac{5 x}{5} \geq \frac{3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{3}{5}$

Divide $x$ by $1$ .

$x \geq \frac{3}{5}$

In the piece where $5 x - 3$ is non-negative, remove the absolute value.

$5 x - 3 < 12$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$5 x - 3 < 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$5 x < 3$

Divide each term in $5 x < 3$ by $5$ and simplify.

Divide each term in $5 x < 3$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

In the piece where $5 x - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 x - 3) < 12$

Write as a piecewise.

${\begin{cases} 5 x - 3 < 12 & x \geq \frac{3}{5} \\ - (5 x - 3) < 12 & x < \frac{3}{5} \end{cases}$

Simplify $- (5 x - 3) < 12$ .

Apply the distributive property.

${\begin{cases} 5 x - 3 < 12 & x \geq \frac{3}{5} \\ - (5 x) - - 3 < 12 & x < \frac{3}{5} \end{cases}$

Multiply $5$ by $- 1$ .

${\begin{cases} 5 x - 3 < 12 & x \geq \frac{3}{5} \\ - 5 x - - 3 < 12 & x < \frac{3}{5} \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} 5 x - 3 < 12 & x \geq \frac{3}{5} \\ - 5 x + 3 < 12 & x < \frac{3}{5} \end{cases}$

Step 2

Solve $5 x - 3 < 12$ when $x \geq \frac{3}{5}$ .

Solve $5 x - 3 < 12$ for $x$ .

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

Add $3$ to both sides of the inequality.

$5 x < 12 + 3$

Add $12$ and $3$ .

$5 x < 15$

Divide each term in $5 x < 15$ by $5$ and simplify.

Divide each term in $5 x < 15$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $15$ by $5$ .

$x < 3$

Find the intersection of $x < 3$ and $x \geq \frac{3}{5}$ .

$\frac{3}{5} \leq x < 3$

Step 3

Solve $- 5 x + 3 < 12$ when $x < \frac{3}{5}$ .

Solve $- 5 x + 3 < 12$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$- 5 x < 12 - 3$

Subtract $3$ from $12$ .

$- 5 x < 9$

Divide each term in $- 5 x < 9$ by $- 5$ and simplify.

Divide each term in $- 5 x < 9$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{9}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{9}{- 5}$

Divide $x$ by $1$ .

$x > \frac{9}{- 5}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{9}{5}$

Find the intersection of $x > - \frac{9}{5}$ and $x < \frac{3}{5}$ .

$- \frac{9}{5} < x < \frac{3}{5}$

Step 4

Find the union of the solutions.

$- \frac{9}{5} < x < 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{9}{5} < x < 3$

Interval Notation:

$(- \frac{9}{5}, 3)$

Step 6

Solve for x |5x-3|<12

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