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Chapter 0: Problem 135

List all numbers that must be excluded from the domain of each rational expression. $$\frac{3}{2 x^{2}+4 x-9}$$

Short Answer

Expert verified
The values to be excluded from the domain of the given rational expression are \(x = 1\) and \(x = -3\).

Step by step solution

01

Identify the Denominator of the Rational Expression

In the problem, the denominator of the rational expression is \(2 x^{2}+4 x-9\). We need to identify the values of \(x\) that will make this expression equal to zero.
02

Set the Denominator Equal to Zero

Establish an equation where the denominator equals zero. This gives us \(2 x^{2}+4 x-9 = 0\). These are the values of \(x\) we are looking for.
03

Solve for \(x\)

Solving quadratic equation yields two values of \(x\). To do this, we can use the quadratic formula, which is \(-b \pm \sqrt{b^{2}-4 a c} \div 2 a\), where \(a\), \(b\) and \(c\) are the coefficients in the quadratic equation. Here \(a = 2\), \(b = 4\) and \(c = -9\) so we plug this into the quadratic formula gives \(x = -1 \pm 2\). Therefore, we have two solutions: \(x = -1 + 2 = 1\) and \(x = -1 - 2 = -3\).
04

Exclude these Values from the Domain

The values of \(x = 1\) and \(x = -3\) will make the denominator zero, we need to exclude these two values from the domain to avoid division by zero.

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Most popular questions from this chapter

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I evaluated \(\frac{3 x-3}{4 x(x-1)}\) for \(x=1\) and obtained 0.

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. The formula $$ I=\frac{1}{4} x+26 $$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$ N=\frac{1}{4} x+6 $$ models the percentage of U.S households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F\), is $$ F=\frac{9}{5} C+32 $$ If Fahrenheit temperature ranges from \(41^{\circ}\) to \(50^{\circ},\) inclusive, what is the range for Celsius temperature? Use interval notation to express this range.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.
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