Solve for x |5x-5|+4<19

$| 5 x - 5 | + 4 < 19$

Step 1

Write $| 5 x - 5 | + 4 < 19$ as a piecewise.

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

$5 x - 5 \geq 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$5 x \geq 5$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $5$ by $5$ .

$x \geq 1$

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

$5 x - 5 + 4 < 19$

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

$5 x - 5 < 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$5 x < 5$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $5$ by $5$ .

$x < 1$

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

$- (5 x - 5) + 4 < 19$

Write as a piecewise.

${\begin{cases} 5 x - 5 + 4 < 19 & x \geq 1 \\ - (5 x - 5) + 4 < 19 & x < 1 \end{cases}$

Add $- 5$ and $4$ .

${\begin{cases} 5 x - 1 < 19 & x \geq 1 \\ - (5 x - 5) + 4 < 19 & x < 1 \end{cases}$

Simplify $- (5 x - 5) + 4 < 19$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x - 1 < 19 & x \geq 1 \\ - (5 x) - - 5 + 4 < 19 & x < 1 \end{cases}$

Multiply $5$ by $- 1$ .

${\begin{cases} 5 x - 1 < 19 & x \geq 1 \\ - 5 x - - 5 + 4 < 19 & x < 1 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} 5 x - 1 < 19 & x \geq 1 \\ - 5 x + 5 + 4 < 19 & x < 1 \end{cases}$

Add $5$ and $4$ .

${\begin{cases} 5 x - 1 < 19 & x \geq 1 \\ - 5 x + 9 < 19 & x < 1 \end{cases}$

Step 2

Solve $5 x - 1 < 19$ when $x \geq 1$ .

Solve $5 x - 1 < 19$ for $x$ .

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

Add $1$ to both sides of the inequality.

$5 x < 19 + 1$

Add $19$ and $1$ .

$5 x < 20$

Divide each term in $5 x < 20$ by $5$ and simplify.

Divide each term in $5 x < 20$ by $5$ .

$\frac{5 x}{5} < \frac{20}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} < \frac{20}{5}$

Divide $x$ by $1$ .

$x < \frac{20}{5}$

Simplify the right side.

Divide $20$ by $5$ .

$x < 4$

Find the intersection of $x < 4$ and $x \geq 1$ .

$1 \leq x < 4$

Step 3

Solve $- 5 x + 9 < 19$ when $x < 1$ .

Solve $- 5 x + 9 < 19$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$- 5 x < 19 - 9$

Subtract $9$ from $19$ .

$- 5 x < 10$

Divide each term in $- 5 x < 10$ by $- 5$ and simplify.

Divide each term in $- 5 x < 10$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{10}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{10}{- 5}$

Divide $x$ by $1$ .

$x > \frac{10}{- 5}$

Simplify the right side.

Divide $10$ by $- 5$ .

$x > - 2$

Find the intersection of $x > - 2$ and $x < 1$ .

$- 2 < x < 1$

Step 4

Find the union of the solutions.

$- 2 < x < 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 4$

Interval Notation:

$(- 2, 4)$

Step 6

Solve for x |5x-5|+4<19

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