GMAT, GRE, SAT Inequalities Quant Question with Equation Manipulation

If 4x > y and y< 0 then which of the following could be the value of x/y ?

A)0
B)1/4
C)1/2
D)1
E)2

The question is asking for what “could be” which means we have multiple answers.

Also, whenever answer choices contain variables or are ratios we can start with plugging in or equation manipulation.

I find that manipulating the equation is easier as it would decrease the amount of work needed if we have to plug in.

Divide by y on both sides

(Usually, you cannot divide inequalities by a variable but since we know here that y < 0 then we can divide by y. Also, since y is negative the sign changes.)

4x > y (dividing by y which we know is negative)
4x/y < 1
x/y < 1/4

We can see that manipulating the equation gave us the answer without the need to plug anything in

A

Refresher: Rules of Inequalities

Inequalities Rule 1

When inequalities are linked up you can skip over the middle inequality.

• If, p < q and q < d, then p < d
• If, p > q and q > d, then p > d

Example: If Brenda is older than Sam and Sam is older than Tim, then Brenda must be older than Tim.

Inequalities Rule 2

Swapping of numbers p and p results in:

• If, p > q, then q < p
• If, p < q, then q < p

Example: Brenda is older than Sam, so Sam is younger than Brenda.

Inequalities Rule 3

Only one of the following is true: p > q or p = q or q > p

Example: Brenda has more money than Sam (a > b). So, Brenda does not have less money than Sam (not p<q). Brenda does not have the same amount of money as Sam (not p=q)

Inequalities Rule 4

Adding of the number d to both sides of inequality If p < q, then p + d < q + d

Example: Brenda has less money than Sam. If both Brenda and Sam get $5 more, then Brenda will still have less money than Sam.

Likewise:

• If p < q, then p − d < q − d
• If p > q, then p + d > q + d, and
• If p > q, then p − d > q − d

So, addition and subtraction of the same value to both p and q will not change the inequality.

Inequalities Rule 5

If you multiply numbers p and q by a positive number, there is no change in inequality. If you multiply both p and q by a negative number, the inequality swaps: p<q becomes q<p after multiplying by (-2)

Here are the rules:

• If p < q, and d is positive, then pd < qd
• If p < q, and d is negative, then pd > qd (inequality swaps)

Positive case example: Brenda’s score of 5 is lower than Sam’s score of 9 (p < q). If Brenda and Sam double their scores ‘×2’, Brenda’s score will still be lower than Sam’s score, 2p < 2q. If the scores turn minuses, then scores will be −p > −q.

Inequalities Rule 6

Putting minuses in front of p and q changes the direction of the inequality.

• If p < q then −p > −q
• If p > q, then −p < −q
• It is the same as multiplying by (-1) and changes direction.

Inequalities Rule 7

Taking the reciprocal 1/value of both p and q changes the direction of the inequality. When a and b are both positive or both negative:

• If, p < q, then1/p > 1/q
• If p > q, then1/p < 1/q

Inequalities Rule 8

A square of a number is always greater than or equal to zero p2≥0 p2≥0

Example: (4)2= 16, (−4)2 = 16, (0)2 = 0

Inequalities Rule 9

Taking a square root will not change the inequality. If p≤q, then √a≤√b, (for p,q≥0) p≤q, then a≤b, (for p,q≥0)

Example:
p=2, q=7
2≤7,√2≤√7

Jeremiah LaBrash is a programmer and CIO for a CCaaS telecom company based in New York, NY. If you have math or verbal questions you’re having difficulty with and would like Jeremiah LaBrash to solve them and parse them into understandable parts, please leave a comment below or mail jr@thelevel11.com