Solve the Inequality for k (4k-1)/3-(2k+1)/7>k

$\frac{4 k - 1}{3} - \frac{2 k + 1}{7} > k$

Step 1

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

To write $\frac{4 k - 1}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{4 k - 1}{3} \cdot \frac{7}{7} - \frac{2 k + 1}{7} > k$

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

$\frac{4 k - 1}{3} \cdot \frac{7}{7} - \frac{2 k + 1}{7} \cdot \frac{3}{3} > k$

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

Multiply $\frac{4 k - 1}{3}$ by $\frac{7}{7}$ .

$\frac{(4 k - 1) \cdot 7}{3 \cdot 7} - \frac{2 k + 1}{7} \cdot \frac{3}{3} > k$

Multiply $3$ by $7$ .

$\frac{(4 k - 1) \cdot 7}{21} - \frac{2 k + 1}{7} \cdot \frac{3}{3} > k$

Multiply $\frac{2 k + 1}{7}$ by $\frac{3}{3}$ .

$\frac{(4 k - 1) \cdot 7}{21} - \frac{(2 k + 1) \cdot 3}{7 \cdot 3} > k$

Multiply $7$ by $3$ .

$\frac{(4 k - 1) \cdot 7}{21} - \frac{(2 k + 1) \cdot 3}{21} > k$

Combine the numerators over the common denominator.

$\frac{(4 k - 1) \cdot 7 - (2 k + 1) \cdot 3}{21} > k$

Simplify the numerator.

Apply the distributive property.

$\frac{4 k \cdot 7 - 1 \cdot 7 - (2 k + 1) \cdot 3}{21} > k$

Multiply $7$ by $4$ .

$\frac{28 k - 1 \cdot 7 - (2 k + 1) \cdot 3}{21} > k$

Multiply $- 1$ by $7$ .

$\frac{28 k - 7 - (2 k + 1) \cdot 3}{21} > k$

Apply the distributive property.

$\frac{28 k - 7 + (- (2 k) - 1 \cdot 1) \cdot 3}{21} > k$

Multiply $2$ by $- 1$ .

$\frac{28 k - 7 + (- 2 k - 1 \cdot 1) \cdot 3}{21} > k$

Multiply $- 1$ by $1$ .

$\frac{28 k - 7 + (- 2 k - 1) \cdot 3}{21} > k$

Apply the distributive property.

$\frac{28 k - 7 - 2 k \cdot 3 - 1 \cdot 3}{21} > k$

Multiply $3$ by $- 2$ .

$\frac{28 k - 7 - 6 k - 1 \cdot 3}{21} > k$

Multiply $- 1$ by $3$ .

$\frac{28 k - 7 - 6 k - 3}{21} > k$

Subtract $6 k$ from $28 k$ .

$\frac{22 k - 7 - 3}{21} > k$

Subtract $3$ from $- 7$ .

$\frac{22 k - 10}{21} > k$

Factor $2$ out of $22 k - 10$ .

Factor $2$ out of $22 k$ .

$\frac{2 (11 k) - 10}{21} > k$

Factor $2$ out of $- 10$ .

$\frac{2 (11 k) + 2 (- 5)}{21} > k$

Factor $2$ out of $2 (11 k) + 2 (- 5)$ .

$\frac{2 (11 k - 5)}{21} > k$

Step 2

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

Subtract $k$ from both sides of the inequality.

$\frac{2 (11 k - 5)}{21} - k > 0$

To write $- k$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$\frac{2 (11 k - 5)}{21} - k \cdot \frac{21}{21} > 0$

Combine $- k$ and $\frac{21}{21}$ .

$\frac{2 (11 k - 5)}{21} + \frac{- k \cdot 21}{21} > 0$

Combine the numerators over the common denominator.

$\frac{2 (11 k - 5) - k \cdot 21}{21} > 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 (11 k) + 2 \cdot - 5 - k \cdot 21}{21} > 0$

Multiply $11$ by $2$ .

$\frac{22 k + 2 \cdot - 5 - k \cdot 21}{21} > 0$

Multiply $2$ by $- 5$ .

$\frac{22 k - 10 - k \cdot 21}{21} > 0$

Multiply $21$ by $- 1$ .

$\frac{22 k - 10 - 21 k}{21} > 0$

Subtract $21 k$ from $22 k$ .

$\frac{k - 10}{21} > 0$

Step 3

Multiply both sides by $21$ .

$\frac{k - 10}{21} \cdot 21 > 0 \cdot 21$

Step 4

Simplify.

Simplify the left side.

Cancel the common factor of $21$ .

Cancel the common factor.

$\frac{k - 10}{21} \cdot 21 > 0 \cdot 21$

Rewrite the expression.

$k - 10 > 0 \cdot 21$

Simplify the right side.

Multiply $0$ by $21$ .

$k - 10 > 0$

Step 5

Add $10$ to both sides of the inequality.

$k > 10$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$k > 10$

Interval Notation:

$(10, \infty)$

Solve the Inequality for k (4k-1)/3-(2k+1)/7>k

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