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Chapter 7: Problem 15

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{y+3}{4 y^{2}+y-3}$$

Short Answer

Expert verified
The rational expression is undefined for \(y=\frac{3}{4}\) and \(y=-1\)

Step by step solution

01

Define the Rational Expression

The given rational expression is \(\frac{y+3}{4y^{2}+y-3}\) where \(y\) is the variable. The denominator of the expression is \(4y^{2}+y-3\).
02

Find the Values that make the Denominator Zero

To find the values of \(y\) that make the denominator zero, we set the denominator equal to zero and solve for \(y\). So, \(4y^{2}+y-3 = 0\).
03

Solve the Quadratic equation

To solve the equation, we look for two numbers that multiply to -12 (4*3) and add up to 1. These numbers are 3 and -4. Consequently, we can factorize the quadratic equation as follows: \(4y^{2}+4y-3y-3 = 0\), obtaining: \(4y(y+1)-3(y+1) = 0\). Hence, the factored form of the equation is: \((4y - 3)(y + 1) = 0\).
04

Solve for the Variable \(y\)

The factored equation \((4y - 3)(y + 1) = 0\) gives us two possible values for \(y\). Setting each factor equal to zero gives the solutions: \(4y - 3 = 0 \Rightarrow y = \frac{3}{4}\) and \(y + 1 = 0 \Rightarrow y = -1\).

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Most popular questions from this chapter

Use a proportion to solve each problem. A person's hair length is proportional to the number of years it has been growing. After 2 years, a person's hair grows 8 inches. The longest moustache on record was grown by Kalyan Sain of India. Sain grew his moustache for 17 years. How long was each side of the moustache?

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

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Will help you prepare for the material covered in the next section. a. If \(y=k x\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

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