Solve the Inequality for x 2x-3<x+2<=3x+5

$2 x - 3 < x + 2 \leq 3 x + 5$

Step 1

Solve $2 x - 3 < x + 2$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $x$ from both sides of the inequality.

$2 x - 3 - x < 2$

Subtract $x$ from $2 x$ .

$x - 3 < 2$

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

Add $3$ to both sides of the inequality.

$x < 2 + 3$

Add $2$ and $3$ .

$x < 5$

Step 2

Solve $x + 2 \leq 3 x + 5$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $3 x$ from both sides of the inequality.

$x + 2 - 3 x \leq 5$

Subtract $3 x$ from $x$ .

$- 2 x + 2 \leq 5$

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

Subtract $2$ from both sides of the inequality.

$- 2 x \leq 5 - 2$

Subtract $2$ from $5$ .

$- 2 x \leq 3$

Divide each term in $- 2 x \leq 3$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{3}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{3}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{3}{2}$

Step 3

Find the intersection of $x < 5$ and $x \geq - \frac{3}{2}$ .

$- \frac{3}{2} \leq x < 5$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{3}{2} \leq x < 5$

Interval Notation:

$[- \frac{3}{2}, 5)$

Step 5

Solve the Inequality for x 2x-3<x+2<=3x+5

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