Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

$x - 3 \geq 0$

Step 1.2

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 1.3

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

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

Step 1.4

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

$x - 3 < 0$

Step 1.5

Add $3$ to both sides of the inequality.

$x < 3$

Step 1.6

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

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

Step 1.7

Write as a piecewise.

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

Step 1.8

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

Tap for more steps...

Step 1.8.1

Simplify each term.

Tap for more steps...

Step 1.8.1.1

Apply the distributive property.

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

Step 1.8.1.2

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

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

Step 1.8.1.3

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

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

Step 1.8.1.4

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Combine the numerators over the common denominator.

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

Step 1.8.5

Simplify the numerator.

Tap for more steps...

Step 1.8.5.1

Multiply $2$ by $4$ .

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

Step 1.8.5.2

Add $- 3$ and $8$ .

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

Step 1.9

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

Tap for more steps...

Step 1.9.1

Simplify each term.

Tap for more steps...

Step 1.9.1.1

Apply the distributive property.

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

Step 1.9.1.2

Multiply $- 1$ by $- 3$ .

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

Step 1.9.1.3

Apply the distributive property.

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

Step 1.9.1.4

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

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

Step 1.9.1.5

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

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

Combine the numerators over the common denominator.

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

Step 1.9.5

Simplify the numerator.

Tap for more steps...

Step 1.9.5.1

Multiply $2$ by $4$ .

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

Step 1.9.5.2

Add $3$ and $8$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Combine the numerators over the common denominator.

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

Step 2.1.1.4

Subtract $5$ from $4$ .

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

Step 2.1.1.5

Move the negative in front of the fraction.

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

Step 2.1.2

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

$x < - 1$

Step 2.2

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

No solution

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Combine the numerators over the common denominator.

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

Step 3.1.1.4

Subtract $11$ from $4$ .

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

Step 3.1.1.5

Move the negative in front of the fraction.

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

Step 3.1.2

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

$- x < - 7$

Step 3.1.3

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

Tap for more steps...

Step 3.1.3.1

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.

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

Step 3.1.3.2

Simplify the left side.

Tap for more steps...

Step 3.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.3.2.2

Divide $x$ by $1$ .

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

Step 3.1.3.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.3.1

Divide $- 7$ by $- 1$ .

$x > 7$

Step 3.2

Find the intersection of $x > 7$ and $x < 3$ .

No solution

Step 4

Find the union of the solutions.

No solution