Precalculus Examples

$\frac{x}{3} + \frac{x}{7} < 2$

Step 1

Simplify $\frac{x}{3} + \frac{x}{7}$ .

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Step 1.1

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

$\frac{x}{3} \cdot \frac{7}{7} + \frac{x}{7} < 2$

Step 1.2

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

$\frac{x}{3} \cdot \frac{7}{7} + \frac{x}{7} \cdot \frac{3}{3} < 2$

Step 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 1.3.1

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

$\frac{x \cdot 7}{3 \cdot 7} + \frac{x}{7} \cdot \frac{3}{3} < 2$

Step 1.3.2

Multiply $3$ by $7$ .

$\frac{x \cdot 7}{21} + \frac{x}{7} \cdot \frac{3}{3} < 2$

Step 1.3.3

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

$\frac{x \cdot 7}{21} + \frac{x \cdot 3}{7 \cdot 3} < 2$

Step 1.3.4

Multiply $7$ by $3$ .

$\frac{x \cdot 7}{21} + \frac{x \cdot 3}{21} < 2$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 7 + x \cdot 3}{21} < 2$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Move $7$ to the left of $x$ .

$\frac{7 \cdot x + x \cdot 3}{21} < 2$

Step 1.5.2

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

$\frac{7 x + 3 \cdot x}{21} < 2$

Step 1.5.3

Add $7 x$ and $3 x$ .

$\frac{10 x}{21} < 2$

Step 2

Multiply both sides by $21$ .

$\frac{10 x}{21} \cdot 21 < 2 \cdot 21$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $21$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{10 x}{21} \cdot 21 < 2 \cdot 21$

Step 3.1.1.2

Rewrite the expression.

$10 x < 2 \cdot 21$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $21$ .

$10 x < 42$

Step 4

Divide each term in $10 x < 42$ by $10$ and simplify.

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Step 4.1

Divide each term in $10 x < 42$ by $10$ .

$\frac{10 x}{10} < \frac{42}{10}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $10$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{10 x}{10} < \frac{42}{10}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x < \frac{42}{10}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $42$ and $10$ .

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Step 4.3.1.1

Factor $2$ out of $42$ .

$x < \frac{2 (21)}{10}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $10$ .

$x < \frac{2 \cdot 21}{2 \cdot 5}$

Step 4.3.1.2.2

Cancel the common factor.

$x < \frac{2 \cdot 21}{2 \cdot 5}$

Step 4.3.1.2.3

Rewrite the expression.

$x < \frac{21}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{21}{5}$

Interval Notation:

$(- \infty, \frac{21}{5})$