Solve the Inequality for x -3x>=-(3x+1)-1/2x

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

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

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

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

Simplify each term.

Apply the distributive property.

$- (3 x) - 1 \cdot 1 - \frac{1}{2} x \leq - 3 x$

Multiply $3$ by $- 1$ .

$- 3 x - 1 \cdot 1 - \frac{1}{2} x \leq - 3 x$

Multiply $- 1$ by $1$ .

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

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

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

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

$- 3 x \cdot \frac{2}{2} - \frac{x}{2} - 1 \leq - 3 x$

Simplify terms.

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

$\frac{- 3 x \cdot 2}{2} - \frac{x}{2} - 1 \leq - 3 x$

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 2 - x}{2} - 1 \leq - 3 x$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $- 3 x \cdot 2 - x$ .

Factor $x$ out of $- 3 x \cdot 2$ .

$\frac{x (- 3 \cdot 2) - x}{2} - 1 \leq - 3 x$

Factor $x$ out of $- x$ .

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

Factor $x$ out of $x (- 3 \cdot 2) + x \cdot - 1$ .

$\frac{x (- 3 \cdot 2 - 1)}{2} - 1 \leq - 3 x$

Multiply $- 3$ by $2$ .

$\frac{x (- 6 - 1)}{2} - 1 \leq - 3 x$

Subtract $1$ from $- 6$ .

$\frac{x \cdot - 7}{2} - 1 \leq - 3 x$

Move $- 7$ to the left of $x$ .

$\frac{- 7 \cdot x}{2} - 1 \leq - 3 x$

Move the negative in front of the fraction.

$- \frac{7 x}{2} - 1 \leq - 3 x$

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

Add $3 x$ to both sides of the inequality.

$- \frac{7 x}{2} - 1 + 3 x \leq 0$

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

$- \frac{7 x}{2} + 3 x \cdot \frac{2}{2} - 1 \leq 0$

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

$- \frac{7 x}{2} + \frac{3 x \cdot 2}{2} - 1 \leq 0$

Combine the numerators over the common denominator.

$\frac{- 7 x + 3 x \cdot 2}{2} - 1 \leq 0$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $- 7 x + 3 x \cdot 2$ .

Factor $x$ out of $- 7 x$ .

$\frac{x \cdot - 7 + 3 x \cdot 2}{2} - 1 \leq 0$

Factor $x$ out of $3 x \cdot 2$ .

$\frac{x \cdot - 7 + x (3 \cdot 2)}{2} - 1 \leq 0$

Factor $x$ out of $x \cdot - 7 + x (3 \cdot 2)$ .

$\frac{x (- 7 + 3 \cdot 2)}{2} - 1 \leq 0$

Multiply $3$ by $2$ .

$\frac{x (- 7 + 6)}{2} - 1 \leq 0$

Add $- 7$ and $6$ .

$\frac{x \cdot - 1}{2} - 1 \leq 0$

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2} - 1 \leq 0$

Move the negative in front of the fraction.

$- \frac{x}{2} - 1 \leq 0$

Add $1$ to both sides of the inequality.

$- \frac{x}{2} \leq 1$

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

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

$\frac{- \frac{x}{2}}{- 1} \geq \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} \geq \frac{1}{- 1}$

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

$\frac{x}{2} \geq \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$\frac{x}{2} \geq - 1$

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq - 1 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq - 1 \cdot 2$

Rewrite the expression.

$x \geq - 1 \cdot 2$

Simplify the right side.

Multiply $- 1$ by $2$ .

$x \geq - 2$

The result can be shown in multiple forms.

Inequality Form:

$x \geq - 2$

Interval Notation:

$[- 2, \infty)$

Solve the Inequality for x -3x>=-(3x+1)-1/2x

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