Solve the Inequality for x (3|x-1|)/4<6

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

Step 1

Multiply both sides by $4$ .

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

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$3 | x - 1 | < 6 \cdot 4$

Simplify the right side.

Multiply $6$ by $4$ .

$3 | x - 1 | < 24$

Step 3

Solve for $x$ .

Divide each term in $3 | x - 1 | < 24$ by $3$ and simplify.

Divide each term in $3 | x - 1 | < 24$ by $3$ .

$\frac{3 | x - 1 |}{3} < \frac{24}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 | x - 1 |}{3} < \frac{24}{3}$

Divide $| x - 1 |$ by $1$ .

$| x - 1 | < \frac{24}{3}$

Simplify the right side.

Divide $24$ by $3$ .

$| x - 1 | < 8$

Write $| x - 1 | < 8$ as a piecewise.

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

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

In the piece where $x - 1$ is non-negative, remove the absolute value.

$x - 1 < 8$

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

$x - 1 < 0$

Add $1$ to both sides of the inequality.

$x < 1$

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

$- (x - 1) < 8$

Write as a piecewise.

${\begin{cases} x - 1 < 8 & x \geq 1 \\ - (x - 1) < 8 & x < 1 \end{cases}$

Simplify $- (x - 1) < 8$ .

Apply the distributive property.

${\begin{cases} x - 1 < 8 & x \geq 1 \\ - x - - 1 < 8 & x < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} x - 1 < 8 & x \geq 1 \\ - x + 1 < 8 & x < 1 \end{cases}$

Solve $x - 1 < 8$ when $x \geq 1$ .

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

Add $1$ to both sides of the inequality.

$x < 8 + 1$

Add $8$ and $1$ .

$x < 9$

Find the intersection of $x < 9$ and $x \geq 1$ .

$1 \leq x < 9$

Solve $- x + 1 < 8$ when $x < 1$ .

Solve $- x + 1 < 8$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- x < 8 - 1$

Subtract $1$ from $8$ .

$- x < 7$

Divide each term in $- x < 7$ by $- 1$ and simplify.

Divide each term in $- x < 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{7}{- 1}$

Divide $x$ by $1$ .

$x > \frac{7}{- 1}$

Simplify the right side.

Divide $7$ by $- 1$ .

$x > - 7$

Find the intersection of $x > - 7$ and $x < 1$ .

$- 7 < x < 1$

Find the union of the solutions.

$- 7 < x < 9$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 7 < x < 9$

Interval Notation:

$(- 7, 9)$

Step 5

Solve the Inequality for x (3|x-1|)/4<6

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