Solve for x -pi/2<x-pi/6<pi/2

$- \frac{π}{2} < x - \frac{π}{6} < \frac{π}{2}$

Step 1

Add $\frac{π}{6}$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$- \frac{π}{2} + \frac{π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Step 2

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{π}{2} \cdot \frac{3}{3} + \frac{π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Step 3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$- \frac{π \cdot 3}{2 \cdot 3} + \frac{π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Multiply $2$ by $3$ .

$- \frac{π \cdot 3}{6} + \frac{π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Step 4

Combine the numerators over the common denominator.

$\frac{- π \cdot 3 + π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Step 5

Simplify the numerator.

Multiply $3$ by $- 1$ .

$\frac{- 3 π + π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Add $- 3 π$ and $π$ .

$\frac{- 2 π}{6} < x < \frac{π}{2} + \frac{π}{6}$

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2 π$ .

$\frac{2 (- π)}{6} < x < \frac{π}{2} + \frac{π}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 (- π)}{2 (3)} < x < \frac{π}{2} + \frac{π}{6}$

Cancel the common factor.

$\frac{2 (- π)}{2 \cdot 3} < x < \frac{π}{2} + \frac{π}{6}$

Rewrite the expression.

$\frac{- π}{3} < x < \frac{π}{2} + \frac{π}{6}$

Move the negative in front of the fraction.

$- \frac{π}{3} < x < \frac{π}{2} + \frac{π}{6}$

Step 7

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{π}{3} < x < \frac{π}{2} \cdot \frac{3}{3} + \frac{π}{6}$

Step 8

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$- \frac{π}{3} < x < \frac{π \cdot 3}{2 \cdot 3} + \frac{π}{6}$

Multiply $2$ by $3$ .

$- \frac{π}{3} < x < \frac{π \cdot 3}{6} + \frac{π}{6}$

Step 9

Combine the numerators over the common denominator.

$- \frac{π}{3} < x < \frac{π \cdot 3 + π}{6}$

Step 10

Simplify the numerator.

Move $3$ to the left of $π$ .

$- \frac{π}{3} < x < \frac{3 \cdot π + π}{6}$

Add $3 π$ and $π$ .

$- \frac{π}{3} < x < \frac{4 π}{6}$

Step 11

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4 π$ .

$- \frac{π}{3} < x < \frac{2 (2 π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- \frac{π}{3} < x < \frac{2 (2 π)}{2 (3)}$

Cancel the common factor.

$- \frac{π}{3} < x < \frac{2 (2 π)}{2 \cdot 3}$

Rewrite the expression.

$- \frac{π}{3} < x < \frac{2 π}{3}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- \frac{π}{3} < x < \frac{2 π}{3}$

Interval Notation:

$(- \frac{π}{3}, \frac{2 π}{3})$

Step 13

Solve for x -pi/2<x-pi/6<pi/2

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