Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Find the common denominator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify each term.

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Step 2.3.1

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

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Multiply $6$ by $1$ .

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

Step 2.3.4

Apply the distributive property.

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

Step 2.3.5

Multiply $2$ by $6$ .

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

Step 2.3.6

Multiply $6$ by $2$ .

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

Step 2.4

Simplify terms.

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Step 2.4.1

Add $2 x$ and $12 x$ .

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

Step 2.4.2

Subtract $5$ from $12$ .

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

Step 2.4.3

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

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Step 2.4.3.1

Factor $7$ out of $14 x$ .

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

Step 2.4.3.2

Factor $7$ out of $7$ .

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

Step 2.4.3.3

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

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

Step 3

Multiply both sides by $2$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

Step 4.1.1.2

Apply the distributive property.

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

Step 4.1.1.3

Multiply.

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Step 4.1.1.3.1

Multiply $2$ by $7$ .

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

Step 4.1.1.3.2

Multiply $7$ by $1$ .

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $2$ by $1$ .

$14 x + 7 \geq 2$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $7$ from both sides of the inequality.

$14 x \geq 2 - 7$

Step 5.1.2

Subtract $7$ from $2$ .

$14 x \geq - 5$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $14$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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