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Chapter 7: Problem 16

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{y+8}{6 y^{2}-y-2}$$

Short Answer

Expert verified
Therefore, the expression is undefined for \(y = 2/3\) and \(y = -1/2\).

Step by step solution

01

Identify the Denominator

The denominator of the rational expression is \(6y^{2} - y - 2\). We will focus on this algebraic equation as we search for the values of y that make it zero.
02

Factor the Denominator

To factor the polynomial in the denominator, look for two numbers that multiply together to give -12 (which is calculated by multiplying 6 and -2) and add up to -1 (the coefficient of y). The two numbers that fulfil this requirement are -4 and 3. We can then express the polynomial as \(6y^{2} - 4y + 3y - 2\), which can be factored into \(2y(3y-2) + 1(3y-2)\), giving us \((3y-2)(2y+1)\).
03

Find the Values for which the Denominator Equals Zero

Setting each term in the factored denominator equal to zero gives us the following two equations: \(3y - 2 = 0\) and \(2y + 1 = 0\). Solving these two equations gives us the values, \(y = 2/3\) and \(y = -1/2\), respectively.

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Most popular questions from this chapter

Two investments have interest rates that differ by \(1 \% .\) An investment for 1 year at the lower rate earns 175 . The same principal invested for a year at the higher rate earns 200 dollars . What are the two interest rates?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{11}{x+7}-\frac{5}{-x-7}$$

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. I added \(\frac{5}{x-7}\) and \(\frac{3}{7-x}\) by first multiplying the second rational expression by \(-1\)

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$
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