Elementary Algebra

1. The amount of the bill is unknown. We let \(b\) represent the total bill. Then Tom's share is \(\dfrac{b}{5}\text{.}\)

2. The dimensions of the room are unknown. We let \(L\) represent the length of bedroom. Then the price per square foot is \(\dfrac{800}{L^2}\text{.}\)

3. The distance and the speed of the liner are unknown. We let \(D\) represent the distance traveled, and \(R\) the speed, or rate. Then the time for voyage is \(\dfrac{D}{R}\text{.}\)

1. Substitute \(\alert{4}\) for \(a\) in the fraction.

   \begin{align*} \dfrac{a^2+1}{a-2} \amp = \dfrac{\alert{4}^2+1}{\alert{4}-2} \amp \amp \blert{\text{Simplify numerator and denominator.}}\\ \amp = \dfrac{17}{2} \end{align*}

2. Because we cannot divide by zero, a fraction is undefined if its denominator is zero. This fraction is undefined for \(a=\alert{2}\text{,}\) because \(~\dfrac{\alert{2}^2+1}{\alert{2}-2}=\dfrac{5}{0}\text{.}\)

We evaluate the average cost for various production levels, \(x\text{,}\) and make a table of values. We plot the points and connect them with a smooth curve to obtain the graph shown below.

\(x\)	\(A\)
\(50\)	\(150\)
\(100\)	\(100\)
\(200\)	\(75\)
\(400\)	\(62.50\)
\(500\)	\(60\)
\(1000\)	\(55\)
\(1250\)	\(54\)
\(2000\)	\(52.50\)

First we consider the numerical part of the fraction: we look for the largest common factor of 42 and 35. This factor is 7, so we write 42 and 35 in factored form:

Next, we write the variable parts of the numerator and denominator in factored form. Remember that \(b^2\) means \(b \cdot b\) and \(b^3\) means \(b \cdot b \cdot b\) .

Finally, we divide any common factors from the numerator and denominator.

The reduced fraction is \(\dfrac{6}{5b}\text{.}\)

We cannot cancel the 4's, because 4 is added to \(8x\text{,}\) not multiplied. Instead, we write the numerator as a product instead of a sum. To do that, we factor the numerator.

We see that 4 is a factor of the numerator, so we can divide top and bottom of the fraction by 4 to get

1. We first factor the numerator and denominator completely.

   \begin{equation*} \dfrac{3x^2+12x}{6x+24} = \dfrac{3x(x+4)}{2 \cdot 3 (x+4)} \end{equation*}

   Then we divide numerator and denominator by any common factors.

   \begin{equation*} \dfrac{3x(x+4)}{2 \cdot 3 (x+4)} = \dfrac{\cancel{3}x\cancel{(x+4)}}{2 \cdot \cancel{3}\cancel{(x+4)}} = \dfrac{x}{2} \end{equation*}

2. If we substitute \(x=\alert{4}\) into the original fraction and simplify, we find

   \begin{equation*} \dfrac{3(\alert{4})^2+12(\alert{4})}{6(\alert{4})+24} = \dfrac{48+48}{24+24} = \dfrac{96}{48} = 2 \end{equation*}

   However, the reduced fraction, \(\dfrac{x}{2}\text{,}\) is equivalent to the original one, so we should get the same answer if we substitute \(x=\alert{4}\) into the reduced fraction:

   \begin{equation*} \dfrac{x}{2} = \dfrac{\alert{4}}{2} = 2 \end{equation*}

1. The opposite of \(~2a-3b~\) is

   \begin{equation*} -(2a-3b) = -2a+3b~~~~ \text{or}~~~~3b-2a \end{equation*}

2. The opposite of \(~-x-1~\) is

   \begin{equation*} -(-x-1) = x+1 \end{equation*}

First, we factor the numerator and denominator.

We see that \(x-2y\) is the opposite of \(2y-x\text{,}\) that is, \(x-2y = -(2y-x)\text{.}\) Thus,