Solve the Inequality for x 2x-3<=4x+3<2x+5

$2 x - 3 \leq 4 x + 3 < 2 x + 5$

Step 1

Solve $2 x - 3 \leq 4 x + 3$ for $x$ .

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

Subtract $4 x$ from both sides of the inequality.

$2 x - 3 - 4 x \leq 3$

Subtract $4 x$ from $2 x$ .

$- 2 x - 3 \leq 3$

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

Add $3$ to both sides of the inequality.

$- 2 x \leq 3 + 3$

Add $3$ and $3$ .

$- 2 x \leq 6$

Divide each term in $- 2 x \leq 6$ by $- 2$ and simplify.

Divide each term in $- 2 x \leq 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{6}{- 2}$

Divide $x$ by $1$ .

$x \geq \frac{6}{- 2}$

Simplify the right side.

Divide $6$ by $- 2$ .

$x \geq - 3$

Step 2

Solve $4 x + 3 < 2 x + 5$ for $x$ .

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

Subtract $2 x$ from both sides of the inequality.

$4 x + 3 - 2 x < 5$

Subtract $2 x$ from $4 x$ .

$2 x + 3 < 5$

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

Subtract $3$ from both sides of the inequality.

$2 x < 5 - 3$

Subtract $3$ from $5$ .

$2 x < 2$

Divide each term in $2 x < 2$ by $2$ and simplify.

Divide each term in $2 x < 2$ by $2$ .

$\frac{2 x}{2} < \frac{2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{2}{2}$

Divide $x$ by $1$ .

$x < \frac{2}{2}$

Simplify the right side.

Divide $2$ by $2$ .

$x < 1$

Step 3

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

$- 3 \leq x < 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x < 1$

Interval Notation:

$[- 3, 1)$

Step 5

Solve the Inequality for x 2x-3<=4x+3<2x+5

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