Solve the Inequality for x 2x-1/2>=7x+7/6

$2 x - \frac{1}{2} \geq 7 x + \frac{7}{6}$

Step 1

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

Subtract $7 x$ from both sides of the inequality.

$2 x - \frac{1}{2} - 7 x \geq \frac{7}{6}$

Subtract $7 x$ from $2 x$ .

$- 5 x - \frac{1}{2} \geq \frac{7}{6}$

Step 2

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

Add $\frac{1}{2}$ to both sides of the inequality.

$- 5 x \geq \frac{7}{6} + \frac{1}{2}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 5 x \geq \frac{7}{6} + \frac{1}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$- 5 x \geq \frac{7}{6} + \frac{3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$- 5 x \geq \frac{7}{6} + \frac{3}{6}$

Combine the numerators over the common denominator.

$- 5 x \geq \frac{7 + 3}{6}$

Add $7$ and $3$ .

$- 5 x \geq \frac{10}{6}$

Cancel the common factor of $10$ and $6$ .

Factor $2$ out of $10$ .

$- 5 x \geq \frac{2 (5)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- 5 x \geq \frac{2 \cdot 5}{2 \cdot 3}$

Cancel the common factor.

$- 5 x \geq \frac{2 \cdot 5}{2 \cdot 3}$

Rewrite the expression.

$- 5 x \geq \frac{5}{3}$

Step 3

Divide each term in $- 5 x \geq \frac{5}{3}$ by $- 5$ and simplify.

Divide each term in $- 5 x \geq \frac{5}{3}$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \leq \frac{\frac{5}{3}}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} \leq \frac{\frac{5}{3}}{- 5}$

Divide $x$ by $1$ .

$x \leq \frac{\frac{5}{3}}{- 5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{5}{3} \cdot \frac{1}{- 5}$

Cancel the common factor of $5$ .

Factor $5$ out of $- 5$ .

$x \leq \frac{5}{3} \cdot \frac{1}{5 (- 1)}$

Cancel the common factor.

$x \leq \frac{5}{3} \cdot \frac{1}{5 \cdot - 1}$

Rewrite the expression.

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

Multiply $\frac{1}{3}$ by $\frac{1}{- 1}$ .

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

Simplify the expression.

Multiply $3$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Step 4

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, - \frac{1}{3}]$

Solve the Inequality for x 2x-1/2>=7x+7/6

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