Algebra Examples

$\frac{4}{5} (4 k - 1) \geq \frac{5}{4} (3 k - 2)$

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 + \frac{4}{5} \cdot (4 k - 1) \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.2

Simplify by adding zeros.

$\frac{4}{5} \cdot (4 k - 1) \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.3

Apply the distributive property.

$\frac{4}{5} (4 k) + \frac{4}{5} \cdot - 1 \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.4

Multiply $\frac{4}{5} (4 k)$ .

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Step 1.4.1

Combine $4$ and $\frac{4}{5}$ .

$\frac{4 \cdot 4}{5} k + \frac{4}{5} \cdot - 1 \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.4.2

Multiply $4$ by $4$ .

$\frac{16}{5} k + \frac{4}{5} \cdot - 1 \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.4.3

Combine $\frac{16}{5}$ and $k$ .

$\frac{16 k}{5} + \frac{4}{5} \cdot - 1 \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.5

Multiply $\frac{4}{5} \cdot - 1$ .

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Step 1.5.1

Combine $\frac{4}{5}$ and $- 1$ .

$\frac{16 k}{5} + \frac{4 \cdot - 1}{5} \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.5.2

Multiply $4$ by $- 1$ .

$\frac{16 k}{5} + \frac{- 4}{5} \geq \frac{5}{4} \cdot (3 k - 2)$

Step 1.6

Move the negative in front of the fraction.

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{5}{4} \cdot (3 k - 2)$

Step 2

Simplify $\frac{5}{4} \cdot (3 k - 2)$ .

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Step 2.1

Apply the distributive property.

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{5}{4} (3 k) + \frac{5}{4} \cdot - 2$

Step 2.2

Multiply $\frac{5}{4} (3 k)$ .

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Step 2.2.1

Combine $3$ and $\frac{5}{4}$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{3 \cdot 5}{4} k + \frac{5}{4} \cdot - 2$

Step 2.2.2

Multiply $3$ by $5$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15}{4} k + \frac{5}{4} \cdot - 2$

Step 2.2.3

Combine $\frac{15}{4}$ and $k$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5}{4} \cdot - 2$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Factor $2$ out of $4$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5}{2 (2)} \cdot - 2$

Step 2.3.2

Factor $2$ out of $- 2$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.3

Cancel the common factor.

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.4

Rewrite the expression.

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5}{2} \cdot - 1$

Step 2.4

Combine $\frac{5}{2}$ and $- 1$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{5 \cdot - 1}{2}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Multiply $5$ by $- 1$ .

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} + \frac{- 5}{2}$

Step 2.5.2

Move the negative in front of the fraction.

$\frac{16 k}{5} - \frac{4}{5} \geq \frac{15 k}{4} - \frac{5}{2}$

Step 3

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

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Step 3.1

Subtract $\frac{15 k}{4}$ from both sides of the inequality.

$\frac{16 k}{5} - \frac{4}{5} - \frac{15 k}{4} \geq - \frac{5}{2}$

Step 3.2

To write $\frac{16 k}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{16 k}{5} \cdot \frac{4}{4} - \frac{15 k}{4} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.3

To write $- \frac{15 k}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{16 k}{5} \cdot \frac{4}{4} - \frac{15 k}{4} \cdot \frac{5}{5} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{16 k}{5}$ by $\frac{4}{4}$ .

$\frac{16 k \cdot 4}{5 \cdot 4} - \frac{15 k}{4} \cdot \frac{5}{5} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.4.2

Multiply $5$ by $4$ .

$\frac{16 k \cdot 4}{20} - \frac{15 k}{4} \cdot \frac{5}{5} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.4.3

Multiply $\frac{15 k}{4}$ by $\frac{5}{5}$ .

$\frac{16 k \cdot 4}{20} - \frac{15 k \cdot 5}{4 \cdot 5} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.4.4

Multiply $4$ by $5$ .

$\frac{16 k \cdot 4}{20} - \frac{15 k \cdot 5}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{16 k \cdot 4 - 15 k \cdot 5}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $k$ out of $16 k \cdot 4 - 15 k \cdot 5$ .

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Step 3.6.1.1.1

Factor $k$ out of $16 k \cdot 4$ .

$\frac{k (16 \cdot 4) - 15 k \cdot 5}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.1.1.2

Factor $k$ out of $- 15 k \cdot 5$ .

$\frac{k (16 \cdot 4) + k (- 15 \cdot 5)}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.1.1.3

Factor $k$ out of $k (16 \cdot 4) + k (- 15 \cdot 5)$ .

$\frac{k (16 \cdot 4 - 15 \cdot 5)}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.1.2

Multiply $16$ by $4$ .

$\frac{k (64 - 15 \cdot 5)}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.1.3

Multiply $- 15$ by $5$ .

$\frac{k (64 - 75)}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.1.4

Subtract $75$ from $64$ .

$\frac{k \cdot - 11}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.2

Move $- 11$ to the left of $k$ .

$\frac{- 11 \cdot k}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{11 k}{20} - \frac{4}{5} \geq - \frac{5}{2}$

Step 4

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

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Step 4.1

Add $\frac{4}{5}$ to both sides of the inequality.

$- \frac{11 k}{20} \geq - \frac{5}{2} + \frac{4}{5}$

Step 4.2

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

$- \frac{11 k}{20} \geq - \frac{5}{2} \cdot \frac{5}{5} + \frac{4}{5}$

Step 4.3

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

$- \frac{11 k}{20} \geq - \frac{5}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2}$

Step 4.4

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

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Step 4.4.1

Multiply $\frac{5}{2}$ by $\frac{5}{5}$ .

$- \frac{11 k}{20} \geq - \frac{5 \cdot 5}{2 \cdot 5} + \frac{4}{5} \cdot \frac{2}{2}$

Step 4.4.2

Multiply $2$ by $5$ .

$- \frac{11 k}{20} \geq - \frac{5 \cdot 5}{10} + \frac{4}{5} \cdot \frac{2}{2}$

Step 4.4.3

Multiply $\frac{4}{5}$ by $\frac{2}{2}$ .

$- \frac{11 k}{20} \geq - \frac{5 \cdot 5}{10} + \frac{4 \cdot 2}{5 \cdot 2}$

Step 4.4.4

Multiply $5$ by $2$ .

$- \frac{11 k}{20} \geq - \frac{5 \cdot 5}{10} + \frac{4 \cdot 2}{10}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{11 k}{20} \geq \frac{- 5 \cdot 5 + 4 \cdot 2}{10}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 5$ by $5$ .

$- \frac{11 k}{20} \geq \frac{- 25 + 4 \cdot 2}{10}$

Step 4.6.2

Multiply $4$ by $2$ .

$- \frac{11 k}{20} \geq \frac{- 25 + 8}{10}$

Step 4.6.3

Add $- 25$ and $8$ .

$- \frac{11 k}{20} \geq \frac{- 17}{10}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{11 k}{20} \geq - \frac{17}{10}$

Step 5

Divide each term in $- \frac{11 k}{20} \geq - \frac{17}{10}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{11 k}{20} \geq - \frac{17}{10}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{11 k}{20}}{- 1} \leq \frac{- \frac{17}{10}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{11 k}{20}}{1} \leq \frac{- \frac{17}{10}}{- 1}$

Step 5.2.2

Divide $\frac{11 k}{20}$ by $1$ .

$\frac{11 k}{20} \leq \frac{- \frac{17}{10}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$\frac{11 k}{20} \leq \frac{\frac{17}{10}}{1}$

Step 5.3.2

Divide $\frac{17}{10}$ by $1$ .

$\frac{11 k}{20} \leq \frac{17}{10}$

Step 6

Multiply both sides by $20$ .

$\frac{11 k}{20} \cdot 20 \leq \frac{17}{10} \cdot 20$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $20$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{11 k}{20} \cdot 20 \leq \frac{17}{10} \cdot 20$

Step 7.1.1.2

Rewrite the expression.

$11 k \leq \frac{17}{10} \cdot 20$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{17}{10} \cdot 20$ .

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Step 7.2.1.1

Cancel the common factor of $10$ .

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Step 7.2.1.1.1

Factor $10$ out of $20$ .

$11 k \leq \frac{17}{10} \cdot (10 (2))$

Step 7.2.1.1.2

Cancel the common factor.

$11 k \leq \frac{17}{10} \cdot (10 \cdot 2)$

Step 7.2.1.1.3

Rewrite the expression.

$11 k \leq 17 \cdot 2$

Step 7.2.1.2

Multiply $17$ by $2$ .

$11 k \leq 34$

Step 8

Divide each term in $11 k \leq 34$ by $11$ and simplify.

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Step 8.1

Divide each term in $11 k \leq 34$ by $11$ .

$\frac{11 k}{11} \leq \frac{34}{11}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $11$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{11 k}{11} \leq \frac{34}{11}$

Step 8.2.1.2

Divide $k$ by $1$ .

$k \leq \frac{34}{11}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$k \leq \frac{34}{11}$

Interval Notation:

$(- \infty, \frac{34}{11}]$