Finite Math Examples

$y = \frac{2 x^{2}}{5 x - 3} - \frac{1}{\sqrt{2 x - 10}}$

Step 1

Set the denominator in $\frac{2 x^{2}}{5 x - 3}$ equal to $0$ to find where the expression is undefined.

$5 x - 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $3$ to both sides of the equation.

$5 x = 3$

Step 2.2

Divide each term in $5 x = 3$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 3

Set the denominator in $\frac{1}{\sqrt{2 x - 10}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2 x - 10} = 0$

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 x - 10}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 10}$ as ${(2 x - 10)}^{\frac{1}{2}}$ .

${({(2 x - 10)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(2 x - 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(2 x - 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x - 10)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(2 x - 10)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(2 x - 10)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$2 x - 10 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$2 x - 10 = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Add $10$ to both sides of the equation.

$2 x = 10$

Step 4.3.2

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 4.3.2.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $10$ by $2$ .

$x = 5$

Step 5

Set the radicand in $\sqrt{2 x - 10}$ less than $0$ to find where the expression is undefined.

$2 x - 10 < 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $10$ to both sides of the inequality.

$2 x < 10$

Step 6.2

Divide each term in $2 x < 10$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 x < 10$ by $2$ .

$\frac{2 x}{2} < \frac{10}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{10}{2}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{10}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $10$ by $2$ .

$x < 5$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 5$

$(- \infty, 5]$

Step 8