Exercise 61 Page 34

We are given the following absolute value equation and we are asked about the number of solutions in two different cases, depending on the values of $a,$ $c,$ and $d.$ Let's begin by isolating the absolute value. Then we can analyze the number of solutions for the different conditions. Now we can consider each case.

Case 1: $a>0\text{ and }c=d$

Notice that the variables $a,$ $d,$ and $c$ appear only on the right-hand side of the equation. The first condition simply tells us that $a$ is a positive number, and the second condition states that $c$ and $d$ are equal. The difference of two equal numbers is always $0.$ We can substitute for $d-c$ into the fraction on the right-hand side and simplify. Let's rewrite the whole equation. An absolute value of an expression equals $0$ only if the expression inside equals $0.$ This means that there is one solution to the equation, since there is only one equation to consider.

Case 2: $a<0\text{ and }c>d$

Again, we will look only at the right-hand side of the equation as this is where we find $a,$ $c,$ and $d.$ We will consider the signs of both the numerator and the denominator of the fraction separately. We instantly know that $a<0$ since this is given in the exercise. Let's now rewrite the second condition $c>d$ so that is matches the form of the expression in the numerator. The difference $d-c$ is less than $0,$ so we know that the numerator is negative. Dividing two negative numbers always gives a positive result. Since the right-hand side of the equation is positive, the equation will have $2$ solutions.