Algebra/Chapter 7/Graphing Systems of Inequalities

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Algebra

Inequalities in 2 variables

A graph of an inequality in 2 variables.

Linear inequalities in 2 variables are typically in the form of $y<mx+b\!\$, where m is the slope of the line and b is the y-intercept.

Graphing an inequality is easy. First, graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b.

Example: Highway and City Gas Mileage

The gas mileage sticker on cars gives two numbers: one for city driving, and one for highway driving. If a sticker says the car gets 25 mpg in the city and 32 mpg on the highway how far can you drive? The abbreviation mpg stands for miles per gallon. The sticker on our car predicts that we get between 25 and 32 mpg, but when we drive our car we estimate how far we are going and how much gas we have in our tank. When we graph we need to change our inequality to a function where x is the number of gallons of gas in our tank. 25x < f(x) < 32x. The picture on the left has two lines one for y=25x and one for y = 32x. The yellow portion of the picture represents how far we may be able to drive when we have x gallons of gas. The vertical line at 10 shows us that we can drive between 250 and 320 miles on 10 gallons of gas. The difference between these two numbers is 70. If we drew a line at 1 gallon of gas we would see that we could drive between 25 and 32 miles, and the difference would be 7. What this graph shows us is the predicted range of miles we can drive on a given amount of gas. The more gas we have, the more likely our actual mileage is going to be different from what the sticker on our car predicted. If our actual mileage falls outside of this range than we may want to take our car to a mechanic to make sure everything is running correctly on our car. If your mileage is too high the odometer on your car may be broken. If your mileage is too low your car might need a tune up.