Solve for x -pi/2<x-pi/4<pi/2

$- \frac{π}{2} < x - \frac{π}{4} < \frac{π}{2}$

Step 1

Add $\frac{π}{4}$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$- \frac{π}{2} + \frac{π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 2

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

$- \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$- \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Multiply $2$ by $2$ .

$- \frac{π \cdot 2}{4} + \frac{π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{- π \cdot 2 + π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 5

Simplify the numerator.

Multiply $2$ by $- 1$ .

$\frac{- 2 π + π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Add $- 2 π$ and $π$ .

$\frac{- π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 6

Move the negative in front of the fraction.

$- \frac{π}{4} < x < \frac{π}{2} + \frac{π}{4}$

Step 7

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

$- \frac{π}{4} < x < \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}$

Step 8

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$- \frac{π}{4} < x < \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}$

Multiply $2$ by $2$ .

$- \frac{π}{4} < x < \frac{π \cdot 2}{4} + \frac{π}{4}$

Step 9

Combine the numerators over the common denominator.

$- \frac{π}{4} < x < \frac{π \cdot 2 + π}{4}$

Step 10

Simplify the numerator.

Move $2$ to the left of $π$ .

$- \frac{π}{4} < x < \frac{2 \cdot π + π}{4}$

Add $2 π$ and $π$ .

$- \frac{π}{4} < x < \frac{3 π}{4}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{π}{4} < x < \frac{3 π}{4}$

Interval Notation:

$(- \frac{π}{4}, \frac{3 π}{4})$

Step 12

Solve for x -pi/2<x-pi/4<pi/2

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