Solve for x 1/4|x-3|+2<1

$\frac{1}{4} | x - 3 | + 2 < 1$

Write $\frac{1}{4} \cdot | x - 3 | + 2 < 1$ as a piecewise.

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

$x - 3 \geq 0$

Add $3$ to both sides of the inequality.

$x \geq 3$

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

$\frac{1}{4} \cdot (x - 3) + 2 < 1$

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

$x - 3 < 0$

Add $3$ to both sides of the inequality.

$x < 3$

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

$\frac{1}{4} \cdot (- (x - 3)) + 2 < 1$

Write as a piecewise.

${\begin{cases} \frac{1}{4} \cdot (x - 3) + 2 < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Simplify $\frac{1}{4} \cdot (x - 3) + 2 < 1$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} \frac{1}{4} x + \frac{1}{4} \cdot - 3 + 2 < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Combine $\frac{1}{4}$ and $x$ .

${\begin{cases} \frac{x}{4} + \frac{1}{4} \cdot - 3 + 2 < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Combine $\frac{1}{4}$ and $- 3$ .

${\begin{cases} \frac{x}{4} + \frac{- 3}{4} + 2 < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Move the negative in front of the fraction.

${\begin{cases} \frac{x}{4} - \frac{3}{4} + 2 < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

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

${\begin{cases} \frac{x}{4} - \frac{3}{4} + 2 \cdot \frac{4}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Combine $2$ and $\frac{4}{4}$ .

${\begin{cases} \frac{x}{4} - \frac{3}{4} + \frac{2 \cdot 4}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Combine the numerators over the common denominator.

${\begin{cases} \frac{x}{4} + \frac{- 3 + 2 \cdot 4}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Simplify the numerator.

Multiply $2$ by $4$ .

${\begin{cases} \frac{x}{4} + \frac{- 3 + 8}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Add $- 3$ and $8$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- (x - 3)) + 2 < 1 & x < 3 \end{cases}$

Simplify $\frac{1}{4} \cdot (- (x - 3)) + 2 < 1$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- x - - 3) + 2 < 1 & x < 3 \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ \frac{1}{4} \cdot (- x + 3) + 2 < 1 & x < 3 \end{cases}$

Apply the distributive property.

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ \frac{1}{4} (- x) + \frac{1}{4} \cdot 3 + 2 < 1 & x < 3 \end{cases}$

Combine $\frac{1}{4}$ and $x$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{1}{4} \cdot 3 + 2 < 1 & x < 3 \end{cases}$

Combine $\frac{1}{4}$ and $3$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{3}{4} + 2 < 1 & x < 3 \end{cases}$

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

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{3}{4} + 2 \cdot \frac{4}{4} < 1 & x < 3 \end{cases}$

Combine $2$ and $\frac{4}{4}$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{3}{4} + \frac{2 \cdot 4}{4} < 1 & x < 3 \end{cases}$

Combine the numerators over the common denominator.

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{3 + 2 \cdot 4}{4} < 1 & x < 3 \end{cases}$

Simplify the numerator.

Multiply $2$ by $4$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{3 + 8}{4} < 1 & x < 3 \end{cases}$

Add $3$ and $8$ .

${\begin{cases} \frac{x}{4} + \frac{5}{4} < 1 & x \geq 3 \\ - \frac{x}{4} + \frac{11}{4} < 1 & x < 3 \end{cases}$

Solve $\frac{x}{4} + \frac{5}{4} < 1$ when $x \geq 3$ .

Solve $\frac{x}{4} + \frac{5}{4} < 1$ for $x$ .

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

Subtract $\frac{5}{4}$ from both sides of the inequality.

$\frac{x}{4} < 1 - \frac{5}{4}$

Write $1$ as a fraction with a common denominator.

$\frac{x}{4} < \frac{4}{4} - \frac{5}{4}$

Combine the numerators over the common denominator.

$\frac{x}{4} < \frac{4 - 5}{4}$

Subtract $5$ from $4$ .

$\frac{x}{4} < \frac{- 1}{4}$

Move the negative in front of the fraction.

$\frac{x}{4} < - \frac{1}{4}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x < - 1$

Find the intersection of $x < - 1$ and $x \geq 3$ .

No solution

Solve $- \frac{x}{4} + \frac{11}{4} < 1$ when $x < 3$ .

Solve $- \frac{x}{4} + \frac{11}{4} < 1$ for $x$ .

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

Subtract $\frac{11}{4}$ from both sides of the inequality.

$- \frac{x}{4} < 1 - \frac{11}{4}$

Write $1$ as a fraction with a common denominator.

$- \frac{x}{4} < \frac{4}{4} - \frac{11}{4}$

Combine the numerators over the common denominator.

$- \frac{x}{4} < \frac{4 - 11}{4}$

Subtract $11$ from $4$ .

$- \frac{x}{4} < \frac{- 7}{4}$

Move the negative in front of the fraction.

$- \frac{x}{4} < - \frac{7}{4}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x < - 7$

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 $x < 3$ .

No solution

Find the union of the solutions.

No solution

Solve for x 1/4|x-3|+2<1

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