2-1 Absolute Value Equations Do the equations |2x + 1| = |3x + 4| and |2x + 1| = 3x + 4 have the same solutions? Explain. A. Yes; squaring both sides of each equation shows that the equations are equivalent. B. Yes; because 3 and 4 are positive numbers, |3x + 4| = 3x + 4. C. No; removing the absolute value symbols from one side of the equation always changes the answer. D. No; one of the solutions of the first equation is an extraneous solution for the second equation.

Both equations are not equal because removing absolute from one sides of the equation changes the answer. You must understand that absolute value of a function can return both positive and negative values. To ascertain this fact, let us solve both equation for x.

For the equation:

|2x + 1| = |3x + 4|

Let's work with the negative values of the absolute function.

-(2x+1) = -(3x+4)

2x+1 = 3x+4

2x-3x = 4-1

-x = 3

x = -3

For the equation |2x + 1| = 3x + 4:

-(2x+1) = 3x+4 (Note that we didn't negate 3x+4 due to the absence of a modulus sign.

Open the parenthesis

-2x-1 = 3x+4

-2x-3x = 4+1

-5x = 5

x = 5/-5

x = -1

We can see that the values of x for both expressions are not the same when considering the negative value of the modulus functions. Hence, removing the absolute value symbols from one side of the equation always changes the answer.