Solve for x (x+3)/6>x/4+1

$\frac{x + 3}{6} > \frac{x}{4} + 1$

Step 1

Multiply both sides by $6$ .

$\frac{x + 3}{6} \cdot 6 > (\frac{x}{4} + 1) \cdot 6$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{x + 3}{6} \cdot 6 > (\frac{x}{4} + 1) \cdot 6$

Rewrite the expression.

$x + 3 > (\frac{x}{4} + 1) \cdot 6$

Simplify the right side.

Simplify $(\frac{x}{4} + 1) \cdot 6$ .

Simplify terms.

Apply the distributive property.

$x + 3 > \frac{x}{4} \cdot 6 + 1 \cdot 6$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$x + 3 > \frac{x}{2 (2)} \cdot 6 + 1 \cdot 6$

Factor $2$ out of $6$ .

$x + 3 > \frac{x}{2 \cdot 2} \cdot (2 \cdot 3) + 1 \cdot 6$

Cancel the common factor.

$x + 3 > \frac{x}{2 \cdot 2} \cdot (2 \cdot 3) + 1 \cdot 6$

Rewrite the expression.

$x + 3 > \frac{x}{2} \cdot 3 + 1 \cdot 6$

Combine $\frac{x}{2}$ and $3$ .

$x + 3 > \frac{x \cdot 3}{2} + 1 \cdot 6$

Multiply $6$ by $1$ .

$x + 3 > \frac{x \cdot 3}{2} + 6$

Move $3$ to the left of $x$ .

$x + 3 > \frac{3 x}{2} + 6$

Step 3

Solve for $x$ .

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

Subtract $\frac{3 x}{2}$ from both sides of the inequality.

$x + 3 - \frac{3 x}{2} > 6$

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} - \frac{3 x}{2} + 3 > 6$

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} - \frac{3 x}{2} + 3 > 6$

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - 3 x}{2} + 3 > 6$

Subtract $3 x$ from $x \cdot 2$ .

Reorder $x$ and $2$ .

$\frac{2 \cdot x - 3 x}{2} + 3 > 6$

Subtract $3 x$ from $2 \cdot x$ .

$\frac{- x}{2} + 3 > 6$

Move the negative in front of the fraction.

$- \frac{x}{2} + 3 > 6$

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

Subtract $3$ from both sides of the inequality.

$- \frac{x}{2} > 6 - 3$

Subtract $3$ from $6$ .

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

Divide each term in $- \frac{x}{2} > 3$ by $- 1$ and simplify.

Divide each term in $- \frac{x}{2} > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $\frac{x}{2}$ by $1$ .

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

Simplify the right side.

Divide $3$ by $- 1$ .

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

Multiply both sides by $2$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x < - 3 \cdot 2$

Simplify the right side.

Multiply $- 3$ by $2$ .

$x < - 6$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < - 6$

Interval Notation:

$(- \infty, - 6)$

Solve for x (x+3)/6>x/4+1

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