Solve the Inequality for x 3+1/2(3-x)<-7

$3 + \frac{1}{2} (3 - x) < - 7$

Step 1

Simplify $3 + \frac{1}{2} \cdot (3 - x)$ .

Simplify each term.

Apply the distributive property.

$3 + \frac{1}{2} \cdot 3 + \frac{1}{2} (- x) < - 7$

Combine $\frac{1}{2}$ and $3$ .

$3 + \frac{3}{2} + \frac{1}{2} (- x) < - 7$

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

$3 + \frac{3}{2} - \frac{x}{2} < - 7$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 \cdot \frac{2}{2} + \frac{3}{2} - \frac{x}{2} < - 7$

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

$\frac{3 \cdot 2}{2} + \frac{3}{2} - \frac{x}{2} < - 7$

Combine the numerators over the common denominator.

$\frac{3 \cdot 2 + 3}{2} - \frac{x}{2} < - 7$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{6 + 3}{2} - \frac{x}{2} < - 7$

Add $6$ and $3$ .

$\frac{9}{2} - \frac{x}{2} < - 7$

Step 2

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

Subtract $\frac{9}{2}$ from both sides of the inequality.

$- \frac{x}{2} < - 7 - \frac{9}{2}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{x}{2} < - 7 \cdot \frac{2}{2} - \frac{9}{2}$

Combine $- 7$ and $\frac{2}{2}$ .

$- \frac{x}{2} < \frac{- 7 \cdot 2}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$- \frac{x}{2} < \frac{- 7 \cdot 2 - 9}{2}$

Simplify the numerator.

Multiply $- 7$ by $2$ .

$- \frac{x}{2} < \frac{- 14 - 9}{2}$

Subtract $9$ from $- 14$ .

$- \frac{x}{2} < \frac{- 23}{2}$

Move the negative in front of the fraction.

$- \frac{x}{2} < - \frac{23}{2}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x < - 23$

Step 4

Divide each term in $- x < - 23$ by $- 1$ and simplify.

Divide each term in $- x < - 23$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 23}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 23}{- 1}$

Divide $x$ by $1$ .

$x > \frac{- 23}{- 1}$

Simplify the right side.

Divide $- 23$ by $- 1$ .

$x > 23$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > 23$

Interval Notation:

$(23, \infty)$

Solve the Inequality for x 3+1/2(3-x)<-7

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