Finite Math Examples

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

Step 1

Subtract $y$ from both sides of the inequality.

$\frac{1}{4 y} - \frac{1}{3} - y < 2$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 y, 3, 1, 1$

Step 2.2

Since $4 y, 3, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $4, 3, 1, 1$ then find LCM for the variable part $y^{1}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 2.5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.7

The LCM of $4, 3, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 3$

Step 2.8

Multiply $2 \cdot 2 \cdot 3$ .

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Step 2.8.1

Multiply $2$ by $2$ .

$4 \cdot 3$

Step 2.8.2

Multiply $4$ by $3$ .

$12$

Step 2.9

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 2.10

The LCM of $y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 2.11

The LCM for $4 y, 3, 1, 1$ is the numeric part $12$ multiplied by the variable part.

$12 y$

Step 3

Multiply each term in $\frac{1}{4 y} - \frac{1}{3} - y < 2$ by $12 y$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{4 y} - \frac{1}{3} - y < 2$ by $12 y$ .

$\frac{1}{4 y} (12 y) - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$12 \frac{1}{4 y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Factor $4$ out of $12$ .

$4 (3) \frac{1}{4 y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.2.2

Factor $4$ out of $4 y$ .

$4 (3) \frac{1}{4 (y)} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.2.3

Cancel the common factor.

$4 \cdot 3 \frac{1}{4 y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.2.4

Rewrite the expression.

$3 \frac{1}{y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.3

Combine $3$ and $\frac{1}{y}$ .

$\frac{3}{y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.4

Cancel the common factor of $y$ .

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Step 3.2.1.4.1

Cancel the common factor.

$\frac{3}{y} y - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.4.2

Rewrite the expression.

$3 - \frac{1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.5

Cancel the common factor of $3$ .

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Step 3.2.1.5.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$3 + \frac{- 1}{3} (12 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.5.2

Factor $3$ out of $12 y$ .

$3 + \frac{- 1}{3} (3 (4 y)) - y (12 y) < 2 (12 y)$

Step 3.2.1.5.3

Cancel the common factor.

$3 + \frac{- 1}{3} (3 (4 y)) - y (12 y) < 2 (12 y)$

Step 3.2.1.5.4

Rewrite the expression.

$3 - (4 y) - y (12 y) < 2 (12 y)$

Step 3.2.1.6

Multiply $4$ by $- 1$ .

$3 - 4 y - y (12 y) < 2 (12 y)$

Step 3.2.1.7

Rewrite using the commutative property of multiplication.

$3 - 4 y - 1 \cdot 12 y \cdot y < 2 (12 y)$

Step 3.2.1.8

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.1.8.1

Move $y$ .

$3 - 4 y - 1 \cdot 12 (y \cdot y) < 2 (12 y)$

Step 3.2.1.8.2

Multiply $y$ by $y$ .

$3 - 4 y - 1 \cdot 12 y^{2} < 2 (12 y)$

Step 3.2.1.9

Multiply $- 1$ by $12$ .

$3 - 4 y - 12 y^{2} < 2 (12 y)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $12$ by $2$ .

$3 - 4 y - 12 y^{2} < 24 y$

Step 4

Solve the inequality.

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Step 4.1

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

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Step 4.1.1

Subtract $24 y$ from both sides of the inequality.

$3 - 4 y - 12 y^{2} - 24 y < 0$

Step 4.1.2

Subtract $24 y$ from $- 4 y$ .

$3 - 12 y^{2} - 28 y < 0$

Step 4.2

Convert the inequality to an equation.

$3 - 12 y^{2} - 28 y = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = - 12$ , $b = - 28$ , and $c = 3$ into the quadratic formula and solve for $y$ .

$\frac{28 \pm \sqrt{{(- 28)}^{2} - 4 \cdot (- 12 \cdot 3)}}{2 \cdot - 12}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 28$ to the power of $2$ .

$y = \frac{28 \pm \sqrt{784 - 4 \cdot - 12 \cdot 3}}{2 \cdot - 12}$

Step 4.5.1.2

Multiply $- 4 \cdot - 12 \cdot 3$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $- 12$ .

$y = \frac{28 \pm \sqrt{784 + 48 \cdot 3}}{2 \cdot - 12}$

Step 4.5.1.2.2

Multiply $48$ by $3$ .

$y = \frac{28 \pm \sqrt{784 + 144}}{2 \cdot - 12}$

Step 4.5.1.3

Add $784$ and $144$ .

$y = \frac{28 \pm \sqrt{928}}{2 \cdot - 12}$

Step 4.5.1.4

Rewrite $928$ as $4^{2} \cdot 58$ .

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Step 4.5.1.4.1

Factor $16$ out of $928$ .

$y = \frac{28 \pm \sqrt{16 (58)}}{2 \cdot - 12}$

Step 4.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{28 \pm \sqrt{4^{2} \cdot 58}}{2 \cdot - 12}$

Step 4.5.1.5

Pull terms out from under the radical.

$y = \frac{28 \pm 4 \sqrt{58}}{2 \cdot - 12}$

Step 4.5.2

Multiply $2$ by $- 12$ .

$y = \frac{28 \pm 4 \sqrt{58}}{- 24}$

Step 4.5.3

Simplify $\frac{28 \pm 4 \sqrt{58}}{- 24}$ .

$y = \frac{7 \pm \sqrt{58}}{- 6}$

Step 4.5.4

Move the negative in front of the fraction.

$y = - \frac{7 \pm \sqrt{58}}{6}$

Step 4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 28$ to the power of $2$ .

$y = \frac{28 \pm \sqrt{784 - 4 \cdot - 12 \cdot 3}}{2 \cdot - 12}$

Step 4.6.1.2

Multiply $- 4 \cdot - 12 \cdot 3$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $- 12$ .

$y = \frac{28 \pm \sqrt{784 + 48 \cdot 3}}{2 \cdot - 12}$

Step 4.6.1.2.2

Multiply $48$ by $3$ .

$y = \frac{28 \pm \sqrt{784 + 144}}{2 \cdot - 12}$

Step 4.6.1.3

Add $784$ and $144$ .

$y = \frac{28 \pm \sqrt{928}}{2 \cdot - 12}$

Step 4.6.1.4

Rewrite $928$ as $4^{2} \cdot 58$ .

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Step 4.6.1.4.1

Factor $16$ out of $928$ .

$y = \frac{28 \pm \sqrt{16 (58)}}{2 \cdot - 12}$

Step 4.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{28 \pm \sqrt{4^{2} \cdot 58}}{2 \cdot - 12}$

Step 4.6.1.5

Pull terms out from under the radical.

$y = \frac{28 \pm 4 \sqrt{58}}{2 \cdot - 12}$

Step 4.6.2

Multiply $2$ by $- 12$ .

$y = \frac{28 \pm 4 \sqrt{58}}{- 24}$

Step 4.6.3

Simplify $\frac{28 \pm 4 \sqrt{58}}{- 24}$ .

$y = \frac{7 \pm \sqrt{58}}{- 6}$

Step 4.6.4

Move the negative in front of the fraction.

$y = - \frac{7 \pm \sqrt{58}}{6}$

Step 4.6.5

Change the $\pm$ to $+$ .

$y = - \frac{7 + \sqrt{58}}{6}$

Step 4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $- 28$ to the power of $2$ .

$y = \frac{28 \pm \sqrt{784 - 4 \cdot - 12 \cdot 3}}{2 \cdot - 12}$

Step 4.7.1.2

Multiply $- 4 \cdot - 12 \cdot 3$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $- 12$ .

$y = \frac{28 \pm \sqrt{784 + 48 \cdot 3}}{2 \cdot - 12}$

Step 4.7.1.2.2

Multiply $48$ by $3$ .

$y = \frac{28 \pm \sqrt{784 + 144}}{2 \cdot - 12}$

Step 4.7.1.3

Add $784$ and $144$ .

$y = \frac{28 \pm \sqrt{928}}{2 \cdot - 12}$

Step 4.7.1.4

Rewrite $928$ as $4^{2} \cdot 58$ .

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Step 4.7.1.4.1

Factor $16$ out of $928$ .

$y = \frac{28 \pm \sqrt{16 (58)}}{2 \cdot - 12}$

Step 4.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{28 \pm \sqrt{4^{2} \cdot 58}}{2 \cdot - 12}$

Step 4.7.1.5

Pull terms out from under the radical.

$y = \frac{28 \pm 4 \sqrt{58}}{2 \cdot - 12}$

Step 4.7.2

Multiply $2$ by $- 12$ .

$y = \frac{28 \pm 4 \sqrt{58}}{- 24}$

Step 4.7.3

Simplify $\frac{28 \pm 4 \sqrt{58}}{- 24}$ .

$y = \frac{7 \pm \sqrt{58}}{- 6}$

Step 4.7.4

Move the negative in front of the fraction.

$y = - \frac{7 \pm \sqrt{58}}{6}$

Step 4.7.5

Change the $\pm$ to $-$ .

$y = - \frac{7 - \sqrt{58}}{6}$

Step 4.8

Consolidate the solutions.

$y = - \frac{7 + \sqrt{58}}{6}, - \frac{7 - \sqrt{58}}{6}$

Step 5

Find the domain of $\frac{1}{4 y} - y - \frac{7}{3}$ .

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Step 5.1

Set the denominator in $\frac{1}{4 y}$ equal to $0$ to find where the expression is undefined.

$4 y = 0$

Step 5.2

Divide each term in $4 y = 0$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 y = 0$ by $4$ .

$\frac{4 y}{4} = \frac{0}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{0}{4}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $4$ .

$y = 0$

Step 5.3

The domain is all values of $y$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 6

Use each root to create test intervals.

$y < - \frac{7 + \sqrt{58}}{6}$

$- \frac{7 + \sqrt{58}}{6} < y < 0$

$0 < y < - \frac{7 - \sqrt{58}}{6}$

$y > - \frac{7 - \sqrt{58}}{6}$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $y < - \frac{7 + \sqrt{58}}{6}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $y < - \frac{7 + \sqrt{58}}{6}$ and see if this value makes the original inequality true.

$y = - 5$

Step 7.1.2

Replace $y$ with $- 5$ in the original inequality.

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

Step 7.1.3

The left side $- 0.38\overline{3}$ is not less than the right side $- 3$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- \frac{7 + \sqrt{58}}{6} < y < 0$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- \frac{7 + \sqrt{58}}{6} < y < 0$ and see if this value makes the original inequality true.

$y = - 1$

Step 7.2.2

Replace $y$ with $- 1$ in the original inequality.

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

Step 7.2.3

The left side $- 0.58\overline{3}$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $0 < y < - \frac{7 - \sqrt{58}}{6}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $0 < y < - \frac{7 - \sqrt{58}}{6}$ and see if this value makes the original inequality true.

$y = 0.05$

Step 7.3.2

Replace $y$ with $0.05$ in the original inequality.

$\frac{1}{4 (0.05)} - \frac{1}{3} < (0.05) + 2$

Step 7.3.3

The left side $4.\overline{6}$ is not less than the right side $2.05$ , which means that the given statement is false.

False

Step 7.4

Test a value on the interval $y > - \frac{7 - \sqrt{58}}{6}$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $y > - \frac{7 - \sqrt{58}}{6}$ and see if this value makes the original inequality true.

$y = 3$

Step 7.4.2

Replace $y$ with $3$ in the original inequality.

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

Step 7.4.3

The left side $- 0.25$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 7.5

Compare the intervals to determine which ones satisfy the original inequality.

$y < - \frac{7 + \sqrt{58}}{6}$ False

$- \frac{7 + \sqrt{58}}{6} < y < 0$ True

$0 < y < - \frac{7 - \sqrt{58}}{6}$ False

$y > - \frac{7 - \sqrt{58}}{6}$ True

$y < - \frac{7 + \sqrt{58}}{6}$ False

$- \frac{7 + \sqrt{58}}{6} < y < 0$ True

$0 < y < - \frac{7 - \sqrt{58}}{6}$ False

$y > - \frac{7 - \sqrt{58}}{6}$ True

Step 8

The solution consists of all of the true intervals.

$- \frac{7 + \sqrt{58}}{6} < y < 0$ or $y > - \frac{7 - \sqrt{58}}{6}$

Step 9

Convert the inequality to interval notation.

$(- \frac{7 + \sqrt{58}}{6}, 0) \cup (- \frac{7 - \sqrt{58}}{6}, \infty)$

Step 10