Question

Solve: $|2x - 1| = |3x + 5|$. write your answers as a list of numbers, separated by commas,

e.g. "23, 19" (but without the quotes).

Answer 1

Solution 1:

If |a| = |b|, then either a = b or a = -b. Hence, if |2x - 1| = |3x + 5|, then either 2x - 1 = 3x + 5 or 2x - 1 = -(3x + 5).

If 2x - 1 = 3x + 5, then x = -6. If 2x - 1 = -(3x + 5), then x = -4/5. Therefore, the solutions are x = \boxed{-6, -4/5}.

Solution 2:

Another approach is to square both sides. Squaring both sides, we get 4x^2 - 4x + 1 = 9x^2 + 30x + 25 (because |a|^2 = a^2 for all a). This simplifies to 5x^2 + 34x + 24 = 0, which factors as (x + 6)(5x + 4) = 0, so the solutions are x = -6 and x = -4/5, as before. You must be careful when using this approach because squaring both sides of an equation can introduce false solutions. Thus, we need to check that both of these "solutions" work in the original expression before declaring them correct.

Answer 2

First, disregard the sign for absolute value and solve for x.

2x - 1 = 3x + 5
2x - 3x = 5 + 1
-x = 6
x = -6

Now, you have to interpret the absolute value. The absolute value of x is always the positive version of whatever the number is. Since x = -6, its absolute value is 6. Therefore, the possible value for x is only one, which is 6.