Flipping the Sign to Inequalities

Inequalities on the GMAT should be approached in the same way as regular equations. We can manipulate inequalities the same way that you can manipulate equations. As with equations on the GMAT, you first must simplify the equation in order to answer the question presented.

The title of this post is: Flipping the sign to inequalities. The only difference to simplifying inequalities compared to normal equations is the times when we have to flip the sign. There are two consistent times when we must always flip the sign:

When we multiple by a negative number.
When we divide by a negative number.

While these situations may seem straight forward, the GMAT has found fun ways to increase the difficulty. Let’s look at an advanced Data Sufficiency question:

1. Is $y > -4$ ?

1) $\left( \frac{1}{7} \right)^{4y} > \left( \frac{1}{7}\right)^{8y + 14}$
2) $4y^2 + 12y < 0[/latex] As we evaluate the first statement, we see that the base of the exponents is the same [latex]\left( \frac{1}{7} \right)[/latex]. Since this inequality is just like an equation, we can drop like bases. However, do you know what happens when you square a fraction? If you square a number between 0 and 1, the number actually gets smaller. Thus, as we look at Statement 1, if we drop the fractional base, we have to flip the sign as well! (To help clarify: [latex]\left( \frac{1}{2} \right)^2 > \left( \frac{1}{2} \right)^3$ . If we calculate the equation, we see the answer is $0.25 > 0.125$ – which is mathematically correct. If we drop the base without flipping the sign, the inequality reads $2 > 3$ , which isn’t mathematically accurate.)

Thus, Statement 1 simplifies to: $4y < 8y + 14[/latex]. As we further simplify the inequality, we move our numbers around and achieve [latex]-14 < 4y[/latex] or [latex]\left( -\frac{14}{4} \right) < y[/latex] or [latex]y > -3.5$ . Since $y > -3.5$ , it is also greater than -4: Sufficient.

Let’s look at statement 2: Quadratics and inequalities are difficult when combined – the squared variable results in two possible solutions. For this example, a savvy test takers notices we can factor out $4y$ from the equation. Translating this to $4y(y+3) < 0[/latex]. As we look at this situation, we know that either [latex]4y[/latex] is negative (less than zero) or [latex](y+3)[/latex] is negative (less than zero) – but not both. Here we probably have to test both situations. Test 1: [latex]4y > 0$ and $(y+3) < 0[/latex] – one positive and one negative (check out the blog post regarding binomials for further reading on the sign issue). If we solve for y in the above situation, we get the outcome of [latex]y>0$ and $y<-3[/latex]. Is this possible? [latex]y[/latex] is both greater than 0 and less then -3? Nope. Thus, this scenario is not the right answer. Test 2: [latex]4y < 0[/latex] and [latex](y+3) > 0$ . If we solve for $y$ in this situation, we get the outcome of $y<0[/latex] and [latex]y>-3$ . Is this possible? Yes! We can re-write this as $-3 < y < 0[/latex]. Or, in other words, Statement 2 is sufficient by itself! The right answer to the above Data Sufficiency question? Either statement on its own is sufficient! Let’s quickly review the times we need to flip the inequality: <ol> <li>When we multiple by a negative number.</li> <li>When we divide by a negative number.</li> <li>When we remove exponential bases that are between 0 and 1</li> <li>When we are dealing with a quadratic equation (there will be two solutions: [latex]x>0$ and $\text{[math]}$ x< 0[/latex]) – don’t forget that binomials usually have two solutions!

Good luck as you practice with inequalities. This was a complicated post! Don’t worry if it doesn’t set in immediately. Print out this screen and read it a couple times – you’ll get it. This is just one of the advanced math topics included among the Kaplan GMAT math material in the newly revised course. If you’ve gotten beyond the intermediate questions and algebra topics, you may be ready to work through advanced topics like this in preparation for the toughest questions on test day.

Brian Fruchey
Kaplan GMAT

Flipping the Sign to Inequalities

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