Solve the Inequality for x 1<=x+6(x+1)-5/2

$1 \leq x + 6 (x + 1) - \frac{5}{2}$

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

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

Simplify $x + 6 (x + 1) - \frac{5}{2}$ .

Find the common denominator.

Write $x$ as a fraction with denominator $1$ .

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

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

$\frac{x}{1} \cdot \frac{2}{2} + 6 (x + 1) - \frac{5}{2} \geq 1$

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

$\frac{x \cdot 2}{2} + 6 (x + 1) - \frac{5}{2} \geq 1$

Write $6 (x + 1)$ as a fraction with denominator $1$ .

$\frac{x \cdot 2}{2} + \frac{6 (x + 1)}{1} - \frac{5}{2} \geq 1$

Multiply $\frac{6 (x + 1)}{1}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{6 (x + 1)}{1} \cdot \frac{2}{2} - \frac{5}{2} \geq 1$

Multiply $\frac{6 (x + 1)}{1}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{6 (x + 1) \cdot 2}{2} - \frac{5}{2} \geq 1$

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + 6 (x + 1) \cdot 2 - 5}{2} \geq 1$

Simplify each term.

Move $2$ to the left of $x$ .

$\frac{2 \cdot x + 6 (x + 1) \cdot 2 - 5}{2} \geq 1$

Apply the distributive property.

$\frac{2 x + (6 x + 6 \cdot 1) \cdot 2 - 5}{2} \geq 1$

Multiply $6$ by $1$ .

$\frac{2 x + (6 x + 6) \cdot 2 - 5}{2} \geq 1$

Apply the distributive property.

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

Multiply $2$ by $6$ .

$\frac{2 x + 12 x + 6 \cdot 2 - 5}{2} \geq 1$

Multiply $6$ by $2$ .

$\frac{2 x + 12 x + 12 - 5}{2} \geq 1$

Simplify terms.

Add $2 x$ and $12 x$ .

$\frac{14 x + 12 - 5}{2} \geq 1$

Subtract $5$ from $12$ .

$\frac{14 x + 7}{2} \geq 1$

Factor $7$ out of $14 x + 7$ .

Factor $7$ out of $14 x$ .

$\frac{7 (2 x) + 7}{2} \geq 1$

Factor $7$ out of $7$ .

$\frac{7 (2 x) + 7 (1)}{2} \geq 1$

Factor $7$ out of $7 (2 x) + 7 (1)$ .

$\frac{7 (2 x + 1)}{2} \geq 1$

Multiply both sides by $2$ .

$\frac{7 (2 x + 1)}{2} \cdot 2 \geq 1 \cdot 2$

Simplify.

Simplify the left side.

Simplify $\frac{7 (2 x + 1)}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{7 (2 x + 1)}{2} \cdot 2 \geq 1 \cdot 2$

Rewrite the expression.

$7 (2 x + 1) \geq 1 \cdot 2$

Apply the distributive property.

$7 (2 x) + 7 \cdot 1 \geq 1 \cdot 2$

Multiply.

Multiply $2$ by $7$ .

$14 x + 7 \cdot 1 \geq 1 \cdot 2$

Multiply $7$ by $1$ .

$14 x + 7 \geq 1 \cdot 2$

Simplify the right side.

Multiply $2$ by $1$ .

$14 x + 7 \geq 2$

Solve for $x$ .

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

Subtract $7$ from both sides of the inequality.

$14 x \geq 2 - 7$

Subtract $7$ from $2$ .

$14 x \geq - 5$

Divide each term in $14 x \geq - 5$ by $14$ and simplify.

Divide each term in $14 x \geq - 5$ by $14$ .

$\frac{14 x}{14} \geq \frac{- 5}{14}$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 x}{14} \geq \frac{- 5}{14}$

Divide $x$ by $1$ .

$x \geq \frac{- 5}{14}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{5}{14}$

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{5}{14}$

Interval Notation:

$[- \frac{5}{14}, \infty)$

Solve the Inequality for x 1<=x+6(x+1)-5/2

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