\displaystyle \left | 2x+1 \right | \gt 5x-4

Which of the following represents the complete solution set for the inequality above?

Question

TheoryPractice · Accepted Answer

\displaystyle x \lt \frac{5}{3}. Whenever you're dealing with absolute values and inequalities, you need to consider two possibilities. In case one, the expression within the absolute value directly satisfies the inequality. In case two, the expression within the absolute value will satisfy the opposite, taken by flipping the inequality sign and negating the entire other side of the inequality (think "a is greater than 1 or less than negative-one" --> case two is that "less than negative..." case). Let's examine both here.

Case 1

\displaystyle 2x+1 \gt 5x-4

Here you can subtract \displaystyle 2x from both sides and add \displaystyle 4 to both sides to isolate the variable:

\displaystyle 5 \gt 3x

Then divide both sides by \displaystyle 3 and you have:

\displaystyle x \lt \frac{5}{3}

It's a good idea to try a number that just barely satisfies your new inequality to ensure that it fits with the given inequality. If you try \displaystyle x=1 here, you'll see that you have a fit: \displaystyle \left | 2(1)+1 \right | \gt 5(1)-4 simplifies to \displaystyle 3 \gt 1, proving that you have a fit.

Case 2

\displaystyle 2x+1 \lt -5x+4 (notice that the sign is flipped, and all terms on the right are negated)

Here you can add \displaystyle 5x to both sides and subtract \displaystyle 1 from both sides to get:

\displaystyle 7x \lt 3

So \displaystyle x \lt \frac{3}{7}

Notice here that this does not add new information. Any number less than \displaystyle \frac{3}{7} is already less than \displaystyle \frac{5}{3} and you've already tried a number between them (1) to ensure that it works. So the full solution set here is just any number less than \displaystyle \frac{5}{3}, making the correct answer \displaystyle x \lt \frac{5}{3}.

<tex>\displaystyle \left | 2x+1 \right | \gt 5x-4</tex><br/><br/>Which of the following represents the complete solution set for the inequality above?

Related questions

Free practice here. Timed mocks when you are ready.