Solve the Inequality for x (3x+1)/7-(2-4x)/3>=(-5x-4)/14+(7x)/6

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

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

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

Factor $2$ out of $2$ .

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $7$ by $3$ .

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

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

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

Multiply $3$ by $7$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $3$ .

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

Multiply $3$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 2$ by $1$ .

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

Multiply $- 2$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $- 2$ by $7$ .

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

Multiply $7$ by $4$ .

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

Add $9 x$ and $28 x$ .

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

Subtract $14$ from $3$ .

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

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

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

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

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

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

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

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

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

Multiply $14$ by $3$ .

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

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

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

Multiply $6$ by $7$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $- 5$ .

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

Multiply $- 4$ by $3$ .

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

Multiply $7$ by $7$ .

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

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

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

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

Factor $2$ out of $34 x$ .

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

Factor $2$ out of $- 12$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $42$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $17$ by $- 1$ .

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

Multiply $- 1$ by $- 6$ .

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

Subtract $17 x$ from $37 x$ .

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

Add $- 11$ and $6$ .

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

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

Factor $5$ out of $20 x$ .

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

Factor $5$ out of $- 5$ .

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

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

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

Multiply both sides by $21$ .

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

Simplify.

Simplify the left side.

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

Cancel the common factor of $21$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply.

Multiply $4$ by $5$ .

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

Multiply $5$ by $- 1$ .

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

Simplify the right side.

Multiply $0$ by $21$ .

$20 x - 5 \geq 0$

Solve for $x$ .

Add $5$ to both sides of the inequality.

$20 x \geq 5$

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

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

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

Simplify the left side.

Cancel the common factor of $20$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

Factor $5$ out of $5$ .

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

Cancel the common factors.

Factor $5$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Solve the Inequality for x (3x+1)/7-(2-4x)/3>=(-5x-4)/14+(7x)/6

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