Algebra Examples

$\frac{2 x + 1}{3} + \frac{3 x - 5}{4} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1

Simplify $\frac{2 x + 1}{3} + \frac{3 x - 5}{4}$ .

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Step 1.1

To write $\frac{2 x + 1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 x + 1}{3} \cdot \frac{4}{4} + \frac{3 x - 5}{4} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.2

To write $\frac{3 x - 5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x + 1}{3} \cdot \frac{4}{4} + \frac{3 x - 5}{4} \cdot \frac{3}{3} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 x + 1}{3}$ by $\frac{4}{4}$ .

$\frac{(2 x + 1) \cdot 4}{3 \cdot 4} + \frac{3 x - 5}{4} \cdot \frac{3}{3} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.3.2

Multiply $3$ by $4$ .

$\frac{(2 x + 1) \cdot 4}{12} + \frac{3 x - 5}{4} \cdot \frac{3}{3} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.3.3

Multiply $\frac{3 x - 5}{4}$ by $\frac{3}{3}$ .

$\frac{(2 x + 1) \cdot 4}{12} + \frac{(3 x - 5) \cdot 3}{4 \cdot 3} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.3.4

Multiply $4$ by $3$ .

$\frac{(2 x + 1) \cdot 4}{12} + \frac{(3 x - 5) \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(2 x + 1) \cdot 4 + (3 x - 5) \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{2 x \cdot 4 + 1 \cdot 4 + (3 x - 5) \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.2

Multiply $4$ by $2$ .

$\frac{8 x + 1 \cdot 4 + (3 x - 5) \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.3

Multiply $4$ by $1$ .

$\frac{8 x + 4 + (3 x - 5) \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.4

Apply the distributive property.

$\frac{8 x + 4 + 3 x \cdot 3 - 5 \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.5

Multiply $3$ by $3$ .

$\frac{8 x + 4 + 9 x - 5 \cdot 3}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.6

Multiply $- 5$ by $3$ .

$\frac{8 x + 4 + 9 x - 15}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.7

Add $8 x$ and $9 x$ .

$\frac{17 x + 4 - 15}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 1.5.8

Subtract $15$ from $4$ .

$\frac{17 x - 11}{12} < \frac{4 x - 3}{5} + \frac{x + 1}{2}$

Step 2

Simplify $\frac{4 x - 3}{5} + \frac{x + 1}{2}$ .

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Step 2.1

To write $\frac{4 x - 3}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{17 x - 11}{12} < \frac{4 x - 3}{5} \cdot \frac{2}{2} + \frac{x + 1}{2}$

Step 2.2

To write $\frac{x + 1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{17 x - 11}{12} < \frac{4 x - 3}{5} \cdot \frac{2}{2} + \frac{x + 1}{2} \cdot \frac{5}{5}$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4 x - 3}{5}$ by $\frac{2}{2}$ .

$\frac{17 x - 11}{12} < \frac{(4 x - 3) \cdot 2}{5 \cdot 2} + \frac{x + 1}{2} \cdot \frac{5}{5}$

Step 2.3.2

Multiply $5$ by $2$ .

$\frac{17 x - 11}{12} < \frac{(4 x - 3) \cdot 2}{10} + \frac{x + 1}{2} \cdot \frac{5}{5}$

Step 2.3.3

Multiply $\frac{x + 1}{2}$ by $\frac{5}{5}$ .

$\frac{17 x - 11}{12} < \frac{(4 x - 3) \cdot 2}{10} + \frac{(x + 1) \cdot 5}{2 \cdot 5}$

Step 2.3.4

Multiply $2$ by $5$ .

$\frac{17 x - 11}{12} < \frac{(4 x - 3) \cdot 2}{10} + \frac{(x + 1) \cdot 5}{10}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{17 x - 11}{12} < \frac{(4 x - 3) \cdot 2 + (x + 1) \cdot 5}{10}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{17 x - 11}{12} < \frac{4 x \cdot 2 - 3 \cdot 2 + (x + 1) \cdot 5}{10}$

Step 2.5.2

Multiply $2$ by $4$ .

$\frac{17 x - 11}{12} < \frac{8 x - 3 \cdot 2 + (x + 1) \cdot 5}{10}$

Step 2.5.3

Multiply $- 3$ by $2$ .

$\frac{17 x - 11}{12} < \frac{8 x - 6 + (x + 1) \cdot 5}{10}$

Step 2.5.4

Apply the distributive property.

$\frac{17 x - 11}{12} < \frac{8 x - 6 + x \cdot 5 + 1 \cdot 5}{10}$

Step 2.5.5

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

$\frac{17 x - 11}{12} < \frac{8 x - 6 + 5 \cdot x + 1 \cdot 5}{10}$

Step 2.5.6

Multiply $5$ by $1$ .

$\frac{17 x - 11}{12} < \frac{8 x - 6 + 5 \cdot x + 5}{10}$

Step 2.5.7

Add $8 x$ and $5 x$ .

$\frac{17 x - 11}{12} < \frac{13 x - 6 + 5}{10}$

Step 2.5.8

Add $- 6$ and $5$ .

$\frac{17 x - 11}{12} < \frac{13 x - 1}{10}$

Step 3

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

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Step 3.1

Subtract $\frac{13 x - 1}{10}$ from both sides of the inequality.

$\frac{17 x - 11}{12} - \frac{13 x - 1}{10} < 0$

Step 3.2

To write $\frac{17 x - 11}{12}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{17 x - 11}{12} \cdot \frac{5}{5} - \frac{13 x - 1}{10} < 0$

Step 3.3

To write $- \frac{13 x - 1}{10}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{17 x - 11}{12} \cdot \frac{5}{5} - \frac{13 x - 1}{10} \cdot \frac{6}{6} < 0$

Step 3.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{17 x - 11}{12}$ by $\frac{5}{5}$ .

$\frac{(17 x - 11) \cdot 5}{12 \cdot 5} - \frac{13 x - 1}{10} \cdot \frac{6}{6} < 0$

Step 3.4.2

Multiply $12$ by $5$ .

$\frac{(17 x - 11) \cdot 5}{60} - \frac{13 x - 1}{10} \cdot \frac{6}{6} < 0$

Step 3.4.3

Multiply $\frac{13 x - 1}{10}$ by $\frac{6}{6}$ .

$\frac{(17 x - 11) \cdot 5}{60} - \frac{(13 x - 1) \cdot 6}{10 \cdot 6} < 0$

Step 3.4.4

Multiply $10$ by $6$ .

$\frac{(17 x - 11) \cdot 5}{60} - \frac{(13 x - 1) \cdot 6}{60} < 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(17 x - 11) \cdot 5 - (13 x - 1) \cdot 6}{60} < 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{17 x \cdot 5 - 11 \cdot 5 - (13 x - 1) \cdot 6}{60} < 0$

Step 3.6.2

Multiply $5$ by $17$ .

$\frac{85 x - 11 \cdot 5 - (13 x - 1) \cdot 6}{60} < 0$

Step 3.6.3

Multiply $- 11$ by $5$ .

$\frac{85 x - 55 - (13 x - 1) \cdot 6}{60} < 0$

Step 3.6.4

Apply the distributive property.

$\frac{85 x - 55 + (- (13 x) - - 1) \cdot 6}{60} < 0$

Step 3.6.5

Multiply $13$ by $- 1$ .

$\frac{85 x - 55 + (- 13 x - - 1) \cdot 6}{60} < 0$

Step 3.6.6

Multiply $- 1$ by $- 1$ .

$\frac{85 x - 55 + (- 13 x + 1) \cdot 6}{60} < 0$

Step 3.6.7

Apply the distributive property.

$\frac{85 x - 55 - 13 x \cdot 6 + 1 \cdot 6}{60} < 0$

Step 3.6.8

Multiply $6$ by $- 13$ .

$\frac{85 x - 55 - 78 x + 1 \cdot 6}{60} < 0$

Step 3.6.9

Multiply $6$ by $1$ .

$\frac{85 x - 55 - 78 x + 6}{60} < 0$

Step 3.6.10

Subtract $78 x$ from $85 x$ .

$\frac{7 x - 55 + 6}{60} < 0$

Step 3.6.11

Add $- 55$ and $6$ .

$\frac{7 x - 49}{60} < 0$

Step 3.6.12

Factor $7$ out of $7 x - 49$ .

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Step 3.6.12.1

Factor $7$ out of $7 x$ .

$\frac{7 (x) - 49}{60} < 0$

Step 3.6.12.2

Factor $7$ out of $- 49$ .

$\frac{7 x + 7 \cdot - 7}{60} < 0$

Step 3.6.12.3

Factor $7$ out of $7 x + 7 \cdot - 7$ .

$\frac{7 (x - 7)}{60} < 0$

Step 4

Multiply both sides by $60$ .

$\frac{7 (x - 7)}{60} \cdot 60 < 0 \cdot 60$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{7 (x - 7)}{60} \cdot 60$ .

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Step 5.1.1.1

Cancel the common factor of $60$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{7 (x - 7)}{60} \cdot 60 < 0 \cdot 60$

Step 5.1.1.1.2

Rewrite the expression.

$7 (x - 7) < 0 \cdot 60$

Step 5.1.1.2

Apply the distributive property.

$7 x + 7 \cdot - 7 < 0 \cdot 60$

Step 5.1.1.3

Multiply $7$ by $- 7$ .

$7 x - 49 < 0 \cdot 60$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $0$ by $60$ .

$7 x - 49 < 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $49$ to both sides of the inequality.

$7 x < 49$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $7$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $49$ by $7$ .

$x < 7$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < 7$

Interval Notation:

$(- \infty, 7)$