Solve the Inequality for x 4x<2x+1<=3x+2

$4 x < 2 x + 1 \leq 3 x + 2$

Solve $4 x < 2 x + 1$ for $x$ .

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

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

$4 x - 2 x < 1$

Subtract $2 x$ from $4 x$ .

$2 x < 1$

Divide each term in $2 x < 1$ by $2$ and simplify.

Divide each term in $2 x < 1$ by $2$ .

$\frac{2 x}{2} < \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{1}{2}$

Divide $x$ by $1$ .

$x < \frac{1}{2}$

Solve $2 x + 1 \leq 3 x + 2$ for $x$ .

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

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

$2 x + 1 - 3 x \leq 2$

Subtract $3 x$ from $2 x$ .

$- x + 1 \leq 2$

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

Subtract $1$ from both sides of the inequality.

$- x \leq 2 - 1$

Subtract $1$ from $2$ .

$- x \leq 1$

Divide each term in $- x \leq 1$ by $- 1$ and simplify.

Divide each term in $- x \leq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{1}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x \geq - 1$

Find the intersection of $x < \frac{1}{2}$ and $x \geq - 1$ .

$- 1 \leq x < \frac{1}{2}$

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x < \frac{1}{2}$

Interval Notation:

$[- 1, \frac{1}{2})$

Solve the Inequality for x 4x<2x+1<=3x+2

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