Solve the System of Inequalities -5<=(4-3x)/2<1

$- 5 \leq \frac{4 - 3 x}{2} < 1$

Step 1

Solve $- 5 \leq \frac{4 - 3 x}{2}$ for $x$ .

Multiply both sides by $2$ .

$- 5 \cdot 2 \leq \frac{4 - 3 x}{2} \cdot 2$

Simplify.

Simplify the left side.

Multiply $- 5$ by $2$ .

$- 10 \leq \frac{4 - 3 x}{2} \cdot 2$

Simplify the right side.

Simplify $\frac{4 - 3 x}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$- 10 \leq \frac{4 - 3 x}{2} \cdot 2$

Rewrite the expression.

$- 10 \leq 4 - 3 x$

Reorder $4$ and $- 3 x$ .

$- 10 \leq - 3 x + 4$

Solve for $x$ .

Rewrite so $x$ is on the left side of the inequality.

$- 3 x + 4 \geq - 10$

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

Subtract $4$ from both sides of the inequality.

$- 3 x \geq - 10 - 4$

Subtract $4$ from $- 10$ .

$- 3 x \geq - 14$

Divide each term in $- 3 x \geq - 14$ by $- 3$ and simplify.

Divide each term in $- 3 x \geq - 14$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- 14}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- 14}{- 3}$

Divide $x$ by $1$ .

$x \leq \frac{- 14}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{14}{3}$

Step 2

Solve $\frac{4 - 3 x}{2} < 1$ for $x$ .

Multiply both sides by $2$ .

$\frac{4 - 3 x}{2} \cdot 2 < 1 \cdot 2$

Simplify.

Simplify the left side.

Simplify $\frac{4 - 3 x}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 - 3 x}{2} \cdot 2 < 1 \cdot 2$

Rewrite the expression.

$4 - 3 x < 1 \cdot 2$

Reorder $4$ and $- 3 x$ .

$- 3 x + 4 < 1 \cdot 2$

Simplify the right side.

Multiply $2$ by $1$ .

$- 3 x + 4 < 2$

Solve for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 3 x < 2 - 4$

Subtract $4$ from $2$ .

$- 3 x < - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{2}{3}$

Step 3

Find the intersection of $x \leq \frac{14}{3}$ and $x > \frac{2}{3}$ .

$\frac{2}{3} < x \leq \frac{14}{3}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$\frac{2}{3} < x \leq \frac{14}{3}$

Interval Notation:

$(\frac{2}{3}, \frac{14}{3}]$

Step 5

Solve the System of Inequalities -5<=(4-3x)/2<1

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