Case 1: 1 solution Case 2: 2 solutions Reasoning: See solution for explanation.
Practice makes perfect
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.
a|x+b|+c=d
Let's begin by isolating the absolute value. Then we can analyze the number of solutions for the different conditions.
Notice that the variables a, d, and c appear only on the right-hand side of the equation.
|x+b|=d-c/a
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.
c = d ⇒ d-c= 0
We can substitute for d-c into the fraction on the right-hand side and simplify.
Let's rewrite the whole equation.
|x+b|=0
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 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.
d-c/a
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.
d-c <0 and a<0 ⇒ d-c/a >0
Since the right-hand side of the equation is positive, the equation will have 2 solutions.
