Algebra Examples

$\frac{x + 5}{18} + \frac{x - 7}{10} < \frac{2}{3}$

Step 1

Simplify $\frac{x + 5}{18} + \frac{x - 7}{10}$ .

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Step 1.1

To write $\frac{x + 5}{18}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x + 5}{18} \cdot \frac{5}{5} + \frac{x - 7}{10} < \frac{2}{3}$

Step 1.2

To write $\frac{x - 7}{10}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{x + 5}{18} \cdot \frac{5}{5} + \frac{x - 7}{10} \cdot \frac{9}{9} < \frac{2}{3}$

Step 1.3

Write each expression with a common denominator of $90$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{x + 5}{18}$ by $\frac{5}{5}$ .

$\frac{(x + 5) \cdot 5}{18 \cdot 5} + \frac{x - 7}{10} \cdot \frac{9}{9} < \frac{2}{3}$

Step 1.3.2

Multiply $18$ by $5$ .

$\frac{(x + 5) \cdot 5}{90} + \frac{x - 7}{10} \cdot \frac{9}{9} < \frac{2}{3}$

Step 1.3.3

Multiply $\frac{x - 7}{10}$ by $\frac{9}{9}$ .

$\frac{(x + 5) \cdot 5}{90} + \frac{(x - 7) \cdot 9}{10 \cdot 9} < \frac{2}{3}$

Step 1.3.4

Multiply $10$ by $9$ .

$\frac{(x + 5) \cdot 5}{90} + \frac{(x - 7) \cdot 9}{90} < \frac{2}{3}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(x + 5) \cdot 5 + (x - 7) \cdot 9}{90} < \frac{2}{3}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{x \cdot 5 + 5 \cdot 5 + (x - 7) \cdot 9}{90} < \frac{2}{3}$

Step 1.5.2

Move $5$ to the left of $x$ .

$\frac{5 \cdot x + 5 \cdot 5 + (x - 7) \cdot 9}{90} < \frac{2}{3}$

Step 1.5.3

Multiply $5$ by $5$ .

$\frac{5 \cdot x + 25 + (x - 7) \cdot 9}{90} < \frac{2}{3}$

Step 1.5.4

Apply the distributive property.

$\frac{5 x + 25 + x \cdot 9 - 7 \cdot 9}{90} < \frac{2}{3}$

Step 1.5.5

Move $9$ to the left of $x$ .

$\frac{5 x + 25 + 9 \cdot x - 7 \cdot 9}{90} < \frac{2}{3}$

Step 1.5.6

Multiply $- 7$ by $9$ .

$\frac{5 x + 25 + 9 \cdot x - 63}{90} < \frac{2}{3}$

Step 1.5.7

Add $5 x$ and $9 x$ .

$\frac{14 x + 25 - 63}{90} < \frac{2}{3}$

Step 1.5.8

Subtract $63$ from $25$ .

$\frac{14 x - 38}{90} < \frac{2}{3}$

Step 1.5.9

Factor $2$ out of $14 x - 38$ .

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Step 1.5.9.1

Factor $2$ out of $14 x$ .

$\frac{2 (7 x) - 38}{90} < \frac{2}{3}$

Step 1.5.9.2

Factor $2$ out of $- 38$ .

$\frac{2 (7 x) + 2 (- 19)}{90} < \frac{2}{3}$

Step 1.5.9.3

Factor $2$ out of $2 (7 x) + 2 (- 19)$ .

$\frac{2 (7 x - 19)}{90} < \frac{2}{3}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $2$ out of $90$ .

$\frac{2 (7 x - 19)}{2 \cdot 45} < \frac{2}{3}$

Step 1.6.2

Cancel the common factor.

$\frac{2 (7 x - 19)}{2 \cdot 45} < \frac{2}{3}$

Step 1.6.3

Rewrite the expression.

$\frac{7 x - 19}{45} < \frac{2}{3}$

Step 2

Multiply both sides by $45$ .

$\frac{7 x - 19}{45} \cdot 45 < \frac{2}{3} \cdot 45$

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 $45$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{7 x - 19}{45} \cdot 45 < \frac{2}{3} \cdot 45$

Step 3.1.1.2

Rewrite the expression.

$7 x - 19 < \frac{2}{3} \cdot 45$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{2}{3} \cdot 45$ .

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Factor $3$ out of $45$ .

$7 x - 19 < \frac{2}{3} \cdot (3 (15))$

Step 3.2.1.1.2

Cancel the common factor.

$7 x - 19 < \frac{2}{3} \cdot (3 \cdot 15)$

Step 3.2.1.1.3

Rewrite the expression.

$7 x - 19 < 2 \cdot 15$

Step 3.2.1.2

Multiply $2$ by $15$ .

$7 x - 19 < 30$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $19$ to both sides of the inequality.

$7 x < 30 + 19$

Step 4.1.2

Add $30$ and $19$ .

$7 x < 49$

Step 4.2

Divide each term in $7 x < 49$ by $7$ and simplify.

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Step 4.2.1

Divide each term in $7 x < 49$ by $7$ .

$\frac{7 x}{7} < \frac{49}{7}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $7$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{49}{7}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{49}{7}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $49$ by $7$ .

$x < 7$

Step 5

Convert the inequality to interval notation.

$(- \infty, 7)$

Step 6