If p is a negative number and 0 < s < |p|, which of the following must

If $p$ is a negative number and $0<s<|p|$, which of the following must also be a negative number?

  1. $(p+s)^2$
  2. $(p-s)^2$
  3. $(s-p)^2$
  4. $p^2-s^2$
  5. $s^2-p^2$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 11 of Section 4 of Practice Test 1. Those questions testing our knowledge about Real Numbers can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

We want to determine which expression must be a negative number and we also see absolute value signs $(|\;\;|)$, which suggests that it will test what we know about Real Numbers. We also have an inequality $(<)$, so the question might also test what we know about Solving Linear Inequalities. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. $0<s<|p|$
  2. We want to know which answer choice MUST be negative

 

Develop a Plan

Quickly looking at the answer choices, we see lots of $2\s$ as exponents. We know that we’re looking for an expression that must be negative, and we might remember that any number raised to the power of $2$ cannot be negative. Raising a number to the power of $2$ is the same thing as multiplying it by itself. Since a positive number multiplied by itself will give a positive number, and a negative number multiplied by itself will give a positive number, then a squared number cannot be negative.

Surveying the answer choices, A, B, and C have their entire expressions raised to the power $2$. So we know that these three CANNOT be negative, so we can start by eliminating A, B, and C. Let’s finish the question by trying to use the inequality relating $s$ and $p$ to determine if D or E is correct.

Solve the Question

Our remaining answer choices all have $s^2$ and $p^2$ in them, and we already know that each of these terms, separately, must be positive because they have an even exponent.

From the inequality, we can see that $p$ has a larger magnitude than $s$ because $p$ is further away from $0$ than $s$ is. And since $p^2$ has two numbers multiplied together ($p$ and another $p$) that are both greater than the two numbers multiplied together in $s^2$ ($s$ and another $s$), $p^2$ will have a larger value than $s^2$. Now choosing between D and E, we know that we will get a negative number if we subtract a bigger number from a smaller number. This matches up with E since $p^2>s^2$.

The correct answer is E, $s^2-p^2$.

Alternatively, if we know that both sides of an inequality are positive, then we can square each side of the inequality without worrying about changing the direction of the inequality sign. Let’s look at our inequality with $s$ and $p$:

$$0<|s|<\;p$$

Well, we know that $|s|$ MUST be positive, because of the absolute value signs. And the $p$ is greater than $0$, so it must be positive too. So let’s go ahead and square this inequality:

$0^2<$ $|s|^2$ $<\;p^2$
$ $ $ $ $ $
$0\;<$ $\;s^2$ $<\;p^2$

Ha! Very clever of us squaring the entire inequality. Plus now we can more clearly see that $p^2$ is definitely greater than $s^2$, so $s^2-p^2$ MUST be negative.

What Did We Learn

Let’s always keep in mind that squared expressions can NEVER have negative values. This helped us to quickly eliminate A, B, and C as answer choices in this question. No better feeling that finding easy ways to cross off wrong answers on the GRE. And we also learned that we can square inequalities if we know that both sides of the inequality must be positive.

 

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