Algebra Examples

$\frac{3 x + 1}{7} - \frac{2 - 4 x}{3} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1

Simplify $\frac{3 x + 1}{7} - \frac{2 - 4 x}{3}$ .

Tap for more steps...

Step 1.1

Factor $2$ out of $2 - 4 x$ .

Tap for more steps...

Step 1.1.1

Factor $2$ out of $2$ .

$\frac{3 x + 1}{7} - \frac{2 (1) - 4 x}{3} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.1.2

Factor $2$ out of $- 4 x$ .

$\frac{3 x + 1}{7} - \frac{2 (1) + 2 (- 2 x)}{3} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.1.3

Factor $2$ out of $2 (1) + 2 (- 2 x)$ .

$\frac{3 x + 1}{7} - \frac{2 (1 - 2 x)}{3} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.2

To write $\frac{3 x + 1}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x + 1}{7} \cdot \frac{3}{3} - \frac{2 (1 - 2 x)}{3} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.3

To write $- \frac{2 (1 - 2 x)}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3 x + 1}{7} \cdot \frac{3}{3} - \frac{2 (1 - 2 x)}{3} \cdot \frac{7}{7} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.4

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

Tap for more steps...

Step 1.4.1

Multiply $\frac{3 x + 1}{7}$ by $\frac{3}{3}$ .

$\frac{(3 x + 1) \cdot 3}{7 \cdot 3} - \frac{2 (1 - 2 x)}{3} \cdot \frac{7}{7} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.4.2

Multiply $7$ by $3$ .

$\frac{(3 x + 1) \cdot 3}{21} - \frac{2 (1 - 2 x)}{3} \cdot \frac{7}{7} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.4.3

Multiply $\frac{2 (1 - 2 x)}{3}$ by $\frac{7}{7}$ .

$\frac{(3 x + 1) \cdot 3}{21} - \frac{2 (1 - 2 x) \cdot 7}{3 \cdot 7} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.4.4

Multiply $3$ by $7$ .

$\frac{(3 x + 1) \cdot 3}{21} - \frac{2 (1 - 2 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{(3 x + 1) \cdot 3 - 2 (1 - 2 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Apply the distributive property.

$\frac{3 x \cdot 3 + 1 \cdot 3 - 2 (1 - 2 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.2

Multiply $3$ by $3$ .

$\frac{9 x + 1 \cdot 3 - 2 (1 - 2 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.3

Multiply $3$ by $1$ .

$\frac{9 x + 3 - 2 (1 - 2 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.4

Apply the distributive property.

$\frac{9 x + 3 + (- 2 \cdot 1 - 2 (- 2 x)) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.5

Multiply $- 2$ by $1$ .

$\frac{9 x + 3 + (- 2 - 2 (- 2 x)) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.6

Multiply $- 2$ by $- 2$ .

$\frac{9 x + 3 + (- 2 + 4 x) \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.7

Apply the distributive property.

$\frac{9 x + 3 - 2 \cdot 7 + 4 x \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.8

Multiply $- 2$ by $7$ .

$\frac{9 x + 3 - 14 + 4 x \cdot 7}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.9

Multiply $7$ by $4$ .

$\frac{9 x + 3 - 14 + 28 x}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.10

Add $9 x$ and $28 x$ .

$\frac{37 x + 3 - 14}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 1.6.11

Subtract $14$ from $3$ .

$\frac{37 x - 11}{21} \geq \frac{- 5 x - 4}{14} + \frac{7 x}{6}$

Step 2

Simplify $\frac{- 5 x - 4}{14} + \frac{7 x}{6}$ .

Tap for more steps...

Step 2.1

To write $\frac{- 5 x - 4}{14}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{37 x - 11}{21} \geq \frac{- 5 x - 4}{14} \cdot \frac{3}{3} + \frac{7 x}{6}$

Step 2.2

To write $\frac{7 x}{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{37 x - 11}{21} \geq \frac{- 5 x - 4}{14} \cdot \frac{3}{3} + \frac{7 x}{6} \cdot \frac{7}{7}$

Step 2.3

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

Multiply $\frac{- 5 x - 4}{14}$ by $\frac{3}{3}$ .

$\frac{37 x - 11}{21} \geq \frac{(- 5 x - 4) \cdot 3}{14 \cdot 3} + \frac{7 x}{6} \cdot \frac{7}{7}$

Step 2.3.2

Multiply $14$ by $3$ .

$\frac{37 x - 11}{21} \geq \frac{(- 5 x - 4) \cdot 3}{42} + \frac{7 x}{6} \cdot \frac{7}{7}$

Step 2.3.3

Multiply $\frac{7 x}{6}$ by $\frac{7}{7}$ .

$\frac{37 x - 11}{21} \geq \frac{(- 5 x - 4) \cdot 3}{42} + \frac{7 x \cdot 7}{6 \cdot 7}$

Step 2.3.4

Multiply $6$ by $7$ .

$\frac{37 x - 11}{21} \geq \frac{(- 5 x - 4) \cdot 3}{42} + \frac{7 x \cdot 7}{42}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{37 x - 11}{21} \geq \frac{(- 5 x - 4) \cdot 3 + 7 x \cdot 7}{42}$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$\frac{37 x - 11}{21} \geq \frac{- 5 x \cdot 3 - 4 \cdot 3 + 7 x \cdot 7}{42}$

Step 2.5.2

Multiply $3$ by $- 5$ .

$\frac{37 x - 11}{21} \geq \frac{- 15 x - 4 \cdot 3 + 7 x \cdot 7}{42}$

Step 2.5.3

Multiply $- 4$ by $3$ .

$\frac{37 x - 11}{21} \geq \frac{- 15 x - 12 + 7 x \cdot 7}{42}$

Step 2.5.4

Multiply $7$ by $7$ .

$\frac{37 x - 11}{21} \geq \frac{- 15 x - 12 + 49 x}{42}$

Step 2.5.5

Add $- 15 x$ and $49 x$ .

$\frac{37 x - 11}{21} \geq \frac{34 x - 12}{42}$

Step 2.5.6

Factor $2$ out of $34 x - 12$ .

Tap for more steps...

Step 2.5.6.1

Factor $2$ out of $34 x$ .

$\frac{37 x - 11}{21} \geq \frac{2 (17 x) - 12}{42}$

Step 2.5.6.2

Factor $2$ out of $- 12$ .

$\frac{37 x - 11}{21} \geq \frac{2 (17 x) + 2 (- 6)}{42}$

Step 2.5.6.3

Factor $2$ out of $2 (17 x) + 2 (- 6)$ .

$\frac{37 x - 11}{21} \geq \frac{2 (17 x - 6)}{42}$

Step 2.6

Cancel the common factors.

Tap for more steps...

Step 2.6.1

Factor $2$ out of $42$ .

$\frac{37 x - 11}{21} \geq \frac{2 (17 x - 6)}{2 \cdot 21}$

Step 2.6.2

Cancel the common factor.

$\frac{37 x - 11}{21} \geq \frac{2 (17 x - 6)}{2 \cdot 21}$

Step 2.6.3

Rewrite the expression.

$\frac{37 x - 11}{21} \geq \frac{17 x - 6}{21}$

Step 3

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

Tap for more steps...

Step 3.1

Subtract $\frac{17 x - 6}{21}$ from both sides of the inequality.

$\frac{37 x - 11}{21} - \frac{17 x - 6}{21} \geq 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{37 x - 11 - (17 x - 6)}{21} \geq 0$

Step 3.3

Simplify each term.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

$\frac{37 x - 11 - (17 x) - - 6}{21} \geq 0$

Step 3.3.2

Multiply $17$ by $- 1$ .

$\frac{37 x - 11 - 17 x - - 6}{21} \geq 0$

Step 3.3.3

Multiply $- 1$ by $- 6$ .

$\frac{37 x - 11 - 17 x + 6}{21} \geq 0$

Step 3.4

Subtract $17 x$ from $37 x$ .

$\frac{20 x - 11 + 6}{21} \geq 0$

Step 3.5

Add $- 11$ and $6$ .

$\frac{20 x - 5}{21} \geq 0$

Step 3.6

Factor $5$ out of $20 x - 5$ .

Tap for more steps...

Step 3.6.1

Factor $5$ out of $20 x$ .

$\frac{5 (4 x) - 5}{21} \geq 0$

Step 3.6.2

Factor $5$ out of $- 5$ .

$\frac{5 (4 x) + 5 (- 1)}{21} \geq 0$

Step 3.6.3

Factor $5$ out of $5 (4 x) + 5 (- 1)$ .

$\frac{5 (4 x - 1)}{21} \geq 0$

Step 4

Multiply both sides by $21$ .

$\frac{5 (4 x - 1)}{21} \cdot 21 \geq 0 \cdot 21$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Simplify $\frac{5 (4 x - 1)}{21} \cdot 21$ .

Tap for more steps...

Step 5.1.1.1

Cancel the common factor of $21$ .

Tap for more steps...

Step 5.1.1.1.1

Cancel the common factor.

$\frac{5 (4 x - 1)}{21} \cdot 21 \geq 0 \cdot 21$

Step 5.1.1.1.2

Rewrite the expression.

$5 (4 x - 1) \geq 0 \cdot 21$

Step 5.1.1.2

Apply the distributive property.

$5 (4 x) + 5 \cdot - 1 \geq 0 \cdot 21$

Step 5.1.1.3

Multiply.

Tap for more steps...

Step 5.1.1.3.1

Multiply $4$ by $5$ .

$20 x + 5 \cdot - 1 \geq 0 \cdot 21$

Step 5.1.1.3.2

Multiply $5$ by $- 1$ .

$20 x - 5 \geq 0 \cdot 21$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Multiply $0$ by $21$ .

$20 x - 5 \geq 0$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Add $5$ to both sides of the inequality.

$20 x \geq 5$

Step 6.2

Divide each term in $20 x \geq 5$ by $20$ and simplify.

Tap for more steps...

Step 6.2.1

Divide each term in $20 x \geq 5$ by $20$ .

$\frac{20 x}{20} \geq \frac{5}{20}$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Cancel the common factor of $20$ .

Tap for more steps...

Step 6.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} \geq \frac{5}{20}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{5}{20}$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Cancel the common factor of $5$ and $20$ .

Tap for more steps...

Step 6.2.3.1.1

Factor $5$ out of $5$ .

$x \geq \frac{5 (1)}{20}$

Step 6.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.2.3.1.2.1

Factor $5$ out of $20$ .

$x \geq \frac{5 \cdot 1}{5 \cdot 4}$

Step 6.2.3.1.2.2

Cancel the common factor.

$x \geq \frac{5 \cdot 1}{5 \cdot 4}$

Step 6.2.3.1.2.3

Rewrite the expression.

$x \geq \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \geq \frac{1}{4}$

Interval Notation:

$[\frac{1}{4}, \infty)$