Finite Math Examples

$y = \frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - x)})}{\frac{0.11}{2}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2425.01 (1 - {(1 + 0.11)}^{- (12 - x)}) = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Simplify.

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Step 1.2.2.1.1

Apply the distributive property.

$2425.01 (1 - {(1 + 0.11)}^{- 1 \cdot 12 - - x}) = 0$

Step 1.2.2.1.2

Multiply $- 1$ by $12$ .

$2425.01 (1 - {(1 + 0.11)}^{- 12 - - x}) = 0$

Step 1.2.2.1.3

Multiply $- - x$ .

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Step 1.2.2.1.3.1

Multiply $- 1$ by $- 1$ .

$2425.01 (1 - {(1 + 0.11)}^{- 12 + 1 x}) = 0$

Step 1.2.2.1.3.2

Multiply $x$ by $1$ .

$2425.01 (1 - {(1 + 0.11)}^{- 12 + x}) = 0$

Step 1.2.2.1.4

Add $1$ and $0.11$ .

$2425.01 (1 - {1.11}^{- 12 + x}) = 0$

Step 1.2.2.2

Divide each term in $2425.01 (1 - {1.11}^{- 12 + x}) = 0$ by $2425.01$ and simplify.

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Step 1.2.2.2.1

Divide each term in $2425.01 (1 - {1.11}^{- 12 + x}) = 0$ by $2425.01$ .

$\frac{2425.01 (1 - {1.11}^{- 12 + x})}{2425.01} = \frac{0}{2425.01}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $2425.01$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{2425.01 (1 - {1.11}^{- 12 + x})}{2425.01} = \frac{0}{2425.01}$

Step 1.2.2.2.2.1.2

Divide $1 - {1.11}^{- 12 + x}$ by $1$ .

$1 - {1.11}^{- 12 + x} = \frac{0}{2425.01}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $0$ by $2425.01$ .

$1 - {1.11}^{- 12 + x} = 0$

Step 1.2.2.3

Subtract $1$ from both sides of the equation.

$- {1.11}^{- 12 + x} = - 1$

Step 1.2.2.4

Divide each term in $- {1.11}^{- 12 + x} = - 1$ by $- 1$ and simplify.

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Step 1.2.2.4.1

Divide each term in $- {1.11}^{- 12 + x} = - 1$ by $- 1$ .

$\frac{- {1.11}^{- 12 + x}}{- 1} = \frac{- 1}{- 1}$

Step 1.2.2.4.2

Simplify the left side.

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Step 1.2.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{{1.11}^{- 12 + x}}{1} = \frac{- 1}{- 1}$

Step 1.2.2.4.2.2

Divide ${1.11}^{- 12 + x}$ by $1$ .

${1.11}^{- 12 + x} = \frac{- 1}{- 1}$

Step 1.2.2.4.3

Simplify the right side.

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Step 1.2.2.4.3.1

Divide $- 1$ by $- 1$ .

${1.11}^{- 12 + x} = 1$

Step 1.2.2.5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({1.11}^{- 12 + x}) = \ln (1)$

Step 1.2.2.6

Expand $\ln ({1.11}^{- 12 + x})$ by moving $- 12 + x$ outside the logarithm.

$(- 12 + x) \ln (1.11) = \ln (1)$

Step 1.2.2.7

Simplify the left side.

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Step 1.2.2.7.1

Apply the distributive property.

$- 12 \ln (1.11) + x \ln (1.11) = \ln (1)$

Step 1.2.2.8

Simplify the right side.

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Step 1.2.2.8.1

The natural logarithm of $1$ is $0$ .

$- 12 \ln (1.11) + x \ln (1.11) = 0$

Step 1.2.2.9

Reorder $- 12 \ln (1.11)$ and $x \ln (1.11)$ .

$x \ln (1.11) - 12 \ln (1.11) = 0$

Step 1.2.2.10

Add $12 \ln (1.11)$ to both sides of the equation.

$x \ln (1.11) = 12 \ln (1.11)$

Step 1.2.2.11

Divide each term in $x \ln (1.11) = 12 \ln (1.11)$ by $\ln (1.11)$ and simplify.

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Step 1.2.2.11.1

Divide each term in $x \ln (1.11) = 12 \ln (1.11)$ by $\ln (1.11)$ .

$\frac{x \ln (1.11)}{\ln (1.11)} = \frac{12 \ln (1.11)}{\ln (1.11)}$

Step 1.2.2.11.2

Simplify the left side.

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Step 1.2.2.11.2.1

Cancel the common factor of $\ln (1.11)$ .

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Step 1.2.2.11.2.1.1

Cancel the common factor.

$\frac{x \ln (1.11)}{\ln (1.11)} = \frac{12 \ln (1.11)}{\ln (1.11)}$

Step 1.2.2.11.2.1.2

Divide $x$ by $1$ .

$x = \frac{12 \ln (1.11)}{\ln (1.11)}$

Step 1.2.2.11.3

Simplify the right side.

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Step 1.2.2.11.3.1

Cancel the common factor of $\ln (1.11)$ .

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Step 1.2.2.11.3.1.1

Cancel the common factor.

$x = \frac{12 \ln (1.11)}{\ln (1.11)}$

Step 1.2.2.11.3.1.2

Divide $12$ by $1$ .

$x = 12$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(12, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - (0))})}{\frac{0.11}{2}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - (0))})}{\frac{0.11}{2}}$

Step 2.2.2

Simplify $\frac{2425.01 (1 - {(1 + 0.11)}^{- (12 - (0))})}{\frac{0.11}{2}}$ .

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Step 2.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$y = 2425.01 (1 - {(1 + 0.11)}^{- (12 - (0))}) \frac{2}{0.11}$

Step 2.2.2.2

Simplify each term.

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Step 2.2.2.2.1

Add $1$ and $0.11$ .

$y = 2425.01 (1 - {1.11}^{- (12 - (0))}) \frac{2}{0.11}$

Step 2.2.2.2.2

Subtract $0$ from $12$ .

$y = 2425.01 (1 - {1.11}^{- 1 \cdot 12}) \frac{2}{0.11}$

Step 2.2.2.2.3

Multiply $- 1$ by $12$ .

$y = 2425.01 (1 - {1.11}^{- 12}) \frac{2}{0.11}$

Step 2.2.2.2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = 2425.01 (1 - \frac{1}{{1.11}^{12}}) \frac{2}{0.11}$

Step 2.2.2.2.5

Raise $1.11$ to the power of $12$ .

$y = 2425.01 (1 - \frac{1}{3.49845059}) \frac{2}{0.11}$

Step 2.2.2.2.6

Divide $1$ by $3.49845059$ .

$y = 2425.01 (1 - 1 \cdot 0.28584082) \frac{2}{0.11}$

Step 2.2.2.2.7

Multiply $- 1$ by $0.28584082$ .

$y = 2425.01 (1 - 0.28584082) \frac{2}{0.11}$

Step 2.2.2.3

Simplify the expression.

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Step 2.2.2.3.1

Subtract $0.28584082$ from $1$ .

$y = 2425.01 \cdot 0.71415917 \frac{2}{0.11}$

Step 2.2.2.3.2

Multiply $2425.01$ by $0.71415917$ .

$y = 1731.84314432 (\frac{2}{0.11})$

Step 2.2.2.3.3

Divide $2$ by $0.11$ .

$y = 1731.84314432 \cdot 18.\overline{18}$

Step 2.2.2.3.4

Multiply $1731.84314432$ by $18.\overline{18}$ .

$y = 31488.0571696$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 31488.0571696)$

Step 3

List the intersections.

x-intercept(s): $(12, 0)$

y-intercept(s): $(0, 31488.0571696)$

Step 4