Solve the System of Inequalities 3>(11+k)/4>=-3

$3 > \frac{11 + k}{4} \geq - 3$

Step 1

Solve $3 > \frac{11 + k}{4}$ for $k$ .

Multiply both sides by $4$ .

$3 \cdot 4 > \frac{11 + k}{4} \cdot 4$

Simplify.

Simplify the left side.

Multiply $3$ by $4$ .

$12 > \frac{11 + k}{4} \cdot 4$

Simplify the right side.

Simplify $\frac{11 + k}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$12 > \frac{11 + k}{4} \cdot 4$

Rewrite the expression.

$12 > 11 + k$

Reorder $11$ and $k$ .

$12 > k + 11$

Solve for $k$ .

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

$k + 11 < 12$

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

Subtract $11$ from both sides of the inequality.

$k < 12 - 11$

Subtract $11$ from $12$ .

$k < 1$

Step 2

Solve $\frac{11 + k}{4} \geq - 3$ for $k$ .

Multiply both sides by $4$ .

$\frac{11 + k}{4} \cdot 4 \geq - 3 \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{11 + k}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{11 + k}{4} \cdot 4 \geq - 3 \cdot 4$

Rewrite the expression.

$11 + k \geq - 3 \cdot 4$

Reorder $11$ and $k$ .

$k + 11 \geq - 3 \cdot 4$

Simplify the right side.

Multiply $- 3$ by $4$ .

$k + 11 \geq - 12$

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

Subtract $11$ from both sides of the inequality.

$k \geq - 12 - 11$

Subtract $11$ from $- 12$ .

$k \geq - 23$

Step 3

Find the intersection of $k < 1$ and $k \geq - 23$ .

$- 23 \leq k < 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 23 \leq k < 1$

Interval Notation:

$[- 23, 1)$

Step 5

Solve the System of Inequalities 3>(11+k)/4>=-3

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