Question

There are three possible cases (or scenarios) for how many solutions that an absolute value equation could have. How many solutions are there for each case? Why are their differences in the number of solutions? Give a mathematical example in your explanation.

Answer 1

There are three possible cases for the number of solutions to an absolute value equation:

One solution: In this case, the absolute value of the expression equals a positive number. For example, the equation |x - 3| = 5 has one solution: x = 8 or x = -2.

Two solutions: In this case, the absolute value of the expression equals zero. For example, the equation |x - 3| = 0 has two solutions: x = 3.

No solution: In this case, the absolute value of the expression equals a negative number. However, the absolute value of any expression is always non-negative, so there can be no solutions. For example, the equation |x - 3| = -2 has no solutions.

The reason why there are differences in the number of solutions is because the absolute value function takes any input and returns a non-negative output. When we set an absolute value expression equal to a number, we are essentially splitting the equation into two parts: one where the expression is positive, and one where it is negative. Depending on the value that the absolute value expression is set equal to, we may get only one of these two parts (the positive part), both of them (the zero part), or none of them (the negative part).

For example, let's consider the absolute value equation |2x - 6| = 4. To solve this equation, we can split it into two cases:

Case 1: 2x - 6 = 4. Solving for "x", we get x = 5.

Case 2: -(2x - 6) = 4. Simplifying, we get -2x + 6 = 4, which gives us x = 1.

Therefore, the equation has two solutions: x = 1 and x = 5.

Answer 2

Absolute value equations can have three possible cases based on the value within the absolute value brackets:

One solution: If the value within the absolute value brackets equals zero, there is only one solution. For example, |x| = 0 has the solution x = 0.

Two solutions: If the value within the absolute value brackets is positive, there are two solutions: one positive and one negative. For example, |x| = 3 has two solutions: x = 3 and x = -3.

No solutions: If the value within the absolute value brackets is negative, there are no solutions. For example, |x| = -2 has no solution because the absolute value of any real number is non-negative.

The differences in the number of solutions depend on the nature of the equation and the value within the absolute value brackets. If the value within the absolute value brackets equals zero, there is only one solution; if it is positive, there are two solutions; and if it is negative, there are no solutions.

For example, consider the absolute value equation |x - 5| = 7. If we subtract 5 from both sides, we get |x - 5| - 5 = 7 - 5, which simplifies to |x - 5| = 2.

Since the value within the absolute value brackets is positive, we know that there are two solutions. We can solve for both solutions by setting x - 5 equal to 2 and -2:

x - 5 = 2 => x = 7 x - 5 = -2 => x = 3

Therefore, the solutions to the absolute value equation |x - 5| = 7 are x = 3 and x = 7.

So to summarize, the number of solutions for an absolute value equation depends on the value within the absolute value brackets and can be one, two or zero, depending on the nature of the equation.