Writing and Solving Inequalities

Question

Pat has x quarters and y dimes, and, in addition, has no more than two dollars. Write inequalities that represent this situation and then graph all points in the coordinate plane that satisfy this condition.

Answer 1

First we want to add coefficients to x and y to represent quarters and dimes in their respective decimal forms. This is easy to do for money because their value is built into our society. We'll balance that with the second part of the first sentence, which I interpreted as less than or equal to two dollars. So algebraically, we have:

.25x + .1y <= 2

We want to get this equation into our standard form (y=mx+b) where m is our slope and b is our y-intercept.

We'll subtract both sides by a quarter of x.

.1y <= -.25x + 2

Then divide both sides by y's coefficient, which is one tenth, leaving us with our standard form and satisfying the first task of our problem.

y <= -2.5x + 20

Now, graphing all of the points that satisfy this condition would be a huge, huge task. A line travels from negative infinity to positive infinity, giving us an infinite number of solutions. So we'll make two points, which is the minimum number of points necessary to create the line.

Our standard form has the y-intercept (b) built into it. The y-coordinate of that is 20, and of course the x-coordinate of this and every other y-intercept is 0.

(0,20) is our first point.

If we input an easy number, like ten, into our x-coordinate, we can work out the y-coordinate that way, giving us our second point. Ten is easy in this case because we can make a -25 from the -2.5 we made when we put the equation into standard form. And integers are easier to work with than decimals.

y<=-2.5(10)+20

y<=-25+20

-5<=-25+20

So our second point is (10,-5).

On the coordinate plane, make sure to draw a solid line extending from (0,20) to (10,-5) [or vice versa] and continuing to negative infinity and positive infinity.

[Note: Again, this is my interpretation of what was asked in the first sentence of the problem. If we should've interpreted 'no more than two dollars' as just less than $2, the entire line should be dotted instead of solid.]

Shade to the left of the line, further driving the point that too many quarters and/or dimes will cause the accumulation of their value to be greater than $2, violating the validity of the equation.

For example: 8 quarters + 10 dimes = $3.00

$3.00 is not less than or equal to $2.00

The point (8,10) is outside of the shading.

4 quarters + 3 dimes = $1.30

$1.30 is less than or equal to $2.00

The point (4,3) is within the shading.

Hope that helps!

Answer 2

The answer is: 0.25x + 0.10y <= 2.00

x - quarters

y - dimes