Solve the Inequality for x 1/2x+3<2x-6

$\frac{1}{2} x + 3 < 2 x - 6$

Step 1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

Factor $x$ out of $x - 2 x \cdot 2$ .

Raise $x$ to the power of $1$ .

$\frac{x^{1} - 2 x \cdot 2}{2} + 3 < - 6$

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $- 2 x \cdot 2$ .

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

Factor $x$ out of $x \cdot 1 + x (- 2 \cdot 2)$ .

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

Multiply $- 2$ by $2$ .

$\frac{x (1 - 4)}{2} + 3 < - 6$

Subtract $4$ from $1$ .

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

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

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

Move the negative in front of the fraction.

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

Step 3

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

Subtract $3$ from both sides of the inequality.

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

Subtract $3$ from $- 6$ .

$- \frac{3 x}{2} < - 9$

Step 4

Divide each term in $- \frac{3 x}{2} < - 9$ by $- 1$ and simplify.

Divide each term in $- \frac{3 x}{2} < - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 x}{2}}{- 1} > \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{3 x}{2}}{1} > \frac{- 9}{- 1}$

Divide $\frac{3 x}{2}$ by $1$ .

$\frac{3 x}{2} > \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$\frac{3 x}{2} > 9$

Step 5

Multiply both sides by $2$ .

$\frac{3 x}{2} \cdot 2 > 9 \cdot 2$

Step 6

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 x}{2} \cdot 2 > 9 \cdot 2$

Rewrite the expression.

$3 x > 9 \cdot 2$

Simplify the right side.

Multiply $9$ by $2$ .

$3 x > 18$

Step 7

Divide each term in $3 x > 18$ by $3$ and simplify.

Divide each term in $3 x > 18$ by $3$ .

$\frac{3 x}{3} > \frac{18}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{18}{3}$

Divide $x$ by $1$ .

$x > \frac{18}{3}$

Simplify the right side.

Divide $18$ by $3$ .

$x > 6$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > 6$

Interval Notation:

$(6, \infty)$

Solve the Inequality for x 1/2x+3<2x-6

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