Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $3$ by $2$ .

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $2$ by $3$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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

Step 1.5.2

Apply the distributive property.

$\frac{2 x + (- x - - 1) \cdot 3}{6} \geq 1$

Step 1.5.3

Multiply $- 1$ by $- 1$ .

$\frac{2 x + (- x + 1) \cdot 3}{6} \geq 1$

Step 1.5.4

Apply the distributive property.

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

Step 1.5.5

Multiply $3$ by $- 1$ .

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

Step 1.5.6

Multiply $3$ by $1$ .

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

Step 1.5.7

Subtract $3 x$ from $2 x$ .

$\frac{- x + 3}{6} \geq 1$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 3}{6} \geq 1$

Step 1.6.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.6.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{x - 3}{6} \geq 1$

Step 2

Divide each term in $- \frac{x - 3}{6} \geq 1$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{x - 3}{6} \geq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 3}{6}}{- 1} \leq \frac{1}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 3}{6}}{1} \leq \frac{1}{- 1}$

Step 2.2.2

Divide $\frac{x - 3}{6}$ by $1$ .

$\frac{x - 3}{6} \leq \frac{1}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $1$ by $- 1$ .

$\frac{x - 3}{6} \leq - 1$

Step 3

Multiply both sides by $6$ .

$\frac{x - 3}{6} \cdot 6 \leq - 1 \cdot 6$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{x - 3}{6} \cdot 6 \leq - 1 \cdot 6$

Step 4.1.1.2

Rewrite the expression.

$x - 3 \leq - 1 \cdot 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 1$ by $6$ .

$x - 3 \leq - 6$

Step 5

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

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Step 5.1

Add $3$ to both sides of the inequality.

$x \leq - 6 + 3$

Step 5.2

Add $- 6$ and $3$ .

$x \leq - 3$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3$

Interval Notation:

$(- \infty, - 3]$