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Absolute Value Equations

The section on Advanced Algebra delves into Absolute Value Equations, offering a comprehensive guide on solving these equations, including the necessity to isolate the absolute value, split the equation, and verify solutions to eliminate extraneous roots.
  • Absolute value equations can have two solutions, akin to quadratic equations, necessitating a split into two scenarios: the expression within the absolute value being equal to a positive or a negative value.
  • When the absolute value is not isolated, it must be isolated before the equation can be split into the 'or' equations for solving.
  • Solving absolute value equations involves setting the inside expression equal to both the positive and negative values of the other side, solving these equations separately, and then checking each solution in the original equation to eliminate extraneous roots.
  • Extraneous solutions, which may arise from the algebraic manipulation, must be checked against the original equation to ensure they are valid solutions.
Introduction to Absolute Value Equations
Solving Basic Absolute Value Equations
Generalizing Absolute Value Equation Solutions
Addressing Extraneous Solutions
Practice and Verification

Q: Why can't the negative values be considered as possible answers to the last practice problem?

A: The key here is that x can be negative, and the expression inside the absolute value can be negative, but the absolute value itself can never be negative. Let's take a look at how this plays out in the two equations you mentioned!

For l1 + 2xl = 4 - x we said that

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

Which gave us

x=1 or x= -5

So there are some negatives involved up to this point. But when we plug those values into the original right side of the equation, 4 - x, we get that

l1 + 2xl = 3 or l1 + 2xl = 9 --->So you see that the absolute value is positive in either case

When we go through the same process with l2x + 5l = x  +1, we get that x = -4 or x = -2. And when we plug those values back in to the original, x + 1, we get

l2x + 5l = -3 or l2x + 5l = -1 ---> but an absolute value can never be negative!

Q: What's an extraneous solution and when do we need to check for them?

A: Extraneous solutions are invalid and do not solve the original equation.

On the GRE, you must check your answers on algebra problems involving squaring or taking roots. Extraneous roots are not considered solutions on the GRE.

Squaring both sides of an equation with radicals makes it possible to introduce extraneous roots as solutions. It is essential to check the answers you find to figure out if they are extraneous.

To check if any of your roots are extraneous, plug each of the roots back in to the original equation. If the root does not solve the original problem, then it is extraneous and is not a one of the solutions.

Here is a link to a Magoosh blog about extraneous solutions that you may find helpful! It is written for the GMAT, but is equally applicable to the GRE:

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