Combining Inequalities

Algebra Level 1

If $-2 < x < 2$ and $-5 < y < 5$ , what is the range of all possible values of $y - x$ ?

-3 < y - x < 7

-7 < y -x < 7

- 7 < y- x < 3

-3 < y - x < 3

-5 < y-x < 5

2 solutions

Rohan Gupta
Apr 29, 2015

The greatest value of $y-x$ is where $y$ is maximized and $x$ is minimized, so $y-x < 5 - (-2) = 7.$

Similarly, the least value of $y-x$ is where $y$ is minimized and $x$ is maximized, so $y-x > -5 - (2) = -7.$

It's obvious that all of the values in-between can be achieved, so the range of possible values is $-7 < y-x < 7.$

Nice approach. Most people try to subtract "vertically" and get $3 < y-x < 3$ .

Chung Kevin - 6 years, 1 month ago

i did not get your point according to my knowledge this solution is for $-2 \leq x \leq 2$ and $-5 \leq y \leq 5$ and in question their is no equality sign in equation

Labeeb Ahmad Mahmood - 5 years, 5 months ago

Min Val for x is -1 and min Val for y is -1. Do the math sherlock

Stefan Maco - 4 years, 8 months ago

The variables need not be integers. For example, we could have $x = 0.5$ . Note also that $-5 < y < 5$ so the min integer value of y us not -1.

In order to find the range, say we want to minimize $y - x$ , then we actually want the minimum of $y$ and the maximum of $x$ . For example, if we set $y = -4.9$ and $x = 1.9$ , we actually end up with $y-x = - 6.8$ .

Chung Kevin - 4 years, 8 months ago

Mary Arans
Aug 5, 2018

min(y - x) = min(y) - max(x) = -5 - 2 = -7. max(y - x) = max(y) - min(x) = 5 - -2 = 7

Combining Inequalities

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