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Chapter 7: Problem 17

List all numbers for which each rational expression is undefined. $$ \frac{a}{3 a-12} $$

Short Answer

Expert verified
Undefined when \(a = 4\).

Step by step solution

01

Identify when the expression is undefined

Rational expressions are undefined when their denominators are zero. In this case, identify the denominator, which is \(3a - 12\). The expression is undefined when this equals zero.
02

Set the denominator equal to zero

Set the denominator \(3a - 12\) equal to zero to find the values of \(a\) that make the expression undefined. This gives us the equation: \[ 3a - 12 = 0 \]
03

Solve for \(a\)

Solve the equation \(3a - 12 = 0\) for \(a\). First, add 12 to both sides:\[ 3a - 12 + 12 = 0 + 12 \]which simplifies to\[ 3a = 12 \]Next, divide both sides by 3:\[ a = \frac{12}{3} \]which simplifies to\[ a = 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

denominator zero
In mathematics, rational expressions become undefined when their denominators are zero. Imagine you have a fraction; the denominator is the bottom part of that fraction. Now, if this denominator equals zero, the whole fraction becomes meaningless, or 'undefined'. For example, let's look at the expression \(\frac{a}{3a-12}\). The denominator here is \(3a-12\). To find when this expression is undefined, we need to solve for when the denominator is zero. This is because division by zero is undefined. In general, always check the denominator of your rational expression. If it can be zero at some value of the variable, the expression won鈥檛 make sense for that value.
solving equations
Solving equations involves finding variable values that make an equation true. It's like solving a puzzle. For our problem, we set the denominator equal to zero to find when the rational expression is undefined. From \(3a - 12 = 0\), our goal is to find \(a\) when this equation holds true. Start by isolating \(\boldsymbol{a}\) on one side. Add 12 to both sides to get \(3a = 12\), then divide by 3 to get \(a = 4\). This tells us that at \(a=4\), the denominator is zero, making the original expression undefined.
rational expressions undefined
A rational expression is undefined at any point where its denominator is zero. For the given example \(\frac{a}{3a-12}\), it鈥檚 undefined when \(3a-12=0\). This is because dividing by zero gives no meaningful or defined result. To find where the denominator is zero, solve the equation \(3a - 12 = 0\). As shown, the solution is \(a = 4\). Therefore, the rational expression \(\frac{a}{3a-12}\) is undefined at \(a = 4\). Remember, always check the denominator in your rational expression. If a value of the variable makes the denominator zero, that value makes the expression undefined.

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