Algebra Examples

$\frac{x - 2}{7} - \frac{x + 3}{3} \leq \frac{5 x}{21}$

Step 1

Multiply both sides by $21$ .

$(\frac{x - 2}{7} - \frac{x + 3}{3}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x - 2}{7} - \frac{x + 3}{3}) \cdot 21$ .

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Step 2.1.1.1

To write $\frac{x - 2}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{x - 2}{7} \cdot \frac{3}{3} - \frac{x + 3}{3}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.2

To write $- \frac{x + 3}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$(\frac{x - 2}{7} \cdot \frac{3}{3} - \frac{x + 3}{3} \cdot \frac{7}{7}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.3

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.3.1

Multiply $\frac{x - 2}{7}$ by $\frac{3}{3}$ .

$(\frac{(x - 2) \cdot 3}{7 \cdot 3} - \frac{x + 3}{3} \cdot \frac{7}{7}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.3.2

Multiply $7$ by $3$ .

$(\frac{(x - 2) \cdot 3}{21} - \frac{x + 3}{3} \cdot \frac{7}{7}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.3.3

Multiply $\frac{x + 3}{3}$ by $\frac{7}{7}$ .

$(\frac{(x - 2) \cdot 3}{21} - \frac{(x + 3) \cdot 7}{3 \cdot 7}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.3.4

Multiply $3$ by $7$ .

$(\frac{(x - 2) \cdot 3}{21} - \frac{(x + 3) \cdot 7}{21}) \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.4

Combine the numerators over the common denominator.

$\frac{(x - 2) \cdot 3 - (x + 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

Apply the distributive property.

$\frac{x \cdot 3 - 2 \cdot 3 - (x + 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.2

Move $3$ to the left of $x$ .

$\frac{3 \cdot x - 2 \cdot 3 - (x + 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.3

Multiply $- 2$ by $3$ .

$\frac{3 \cdot x - 6 - (x + 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.4

Apply the distributive property.

$\frac{3 x - 6 + (- x - 1 \cdot 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.5

Multiply $- 1$ by $3$ .

$\frac{3 x - 6 + (- x - 3) \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.6

Apply the distributive property.

$\frac{3 x - 6 - x \cdot 7 - 3 \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.7

Multiply $7$ by $- 1$ .

$\frac{3 x - 6 - 7 x - 3 \cdot 7}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.8

Multiply $- 3$ by $7$ .

$\frac{3 x - 6 - 7 x - 21}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.9

Subtract $7 x$ from $3 x$ .

$\frac{- 4 x - 6 - 21}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.5.10

Subtract $21$ from $- 6$ .

$\frac{- 4 x - 27}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.6

Cancel the common factor of $21$ .

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Step 2.1.1.6.1

Cancel the common factor.

$\frac{- 4 x - 27}{21} \cdot 21 \leq \frac{5 x}{21} \cdot 21$

Step 2.1.1.6.2

Rewrite the expression.

$- 4 x - 27 \leq \frac{5 x}{21} \cdot 21$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $21$ .

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Step 2.2.1.1

Cancel the common factor.

$- 4 x - 27 \leq \frac{5 x}{21} \cdot 21$

Step 2.2.1.2

Rewrite the expression.

$- 4 x - 27 \leq 5 x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $5 x$ from both sides of the inequality.

$- 4 x - 27 - 5 x \leq 0$

Step 3.1.2

Subtract $5 x$ from $- 4 x$ .

$- 9 x - 27 \leq 0$

Step 3.2

Add $27$ to both sides of the inequality.

$- 9 x \leq 27$

Step 3.3

Divide each term in $- 9 x \leq 27$ by $- 9$ and simplify.

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Step 3.3.1

Divide each term in $- 9 x \leq 27$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} \geq \frac{27}{- 9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} \geq \frac{27}{- 9}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{27}{- 9}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $27$ by $- 9$ .

$x \geq - 3$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq - 3$

Interval Notation:

$[- 3, \infty)$