How to Solve & Graph a Solution Set

Finding and graphing solution sets is the core of algebra and calculus.
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If you have an equation ​y​ = ​f​(​x​)​, its solution set is the collection of ​x​ and ​y​ values – often written in the form (​x​, ​y​) – that make the equation true. In other words, they make the right and left sides of the equation equal to each other. Depending on the type of equation you're dealing with, the solution set might be a few points or a line, or it might also be an inequality – all of which you can graph once you've identified two or more points in the solution set.

The Strategy for Identifying Your Solution Set

Identifying the solution set of an equation usually involves three steps: First, you solve the equation for one variable in terms of the other; the convention is to solve for ​y​ in terms of ​x.​ Next, you identify which ​x​ values can be part of your solution set. And finally, you substitute ​x​ values into the equation to find the corresponding ​y​ values.

Tips

  • If you've been asked to graph your solution set, you don't have to find every single point in it. You only need enough to define the line formed by the solution set.

Example 1.​ Solve for the solution set of

2y = 6x

    What "solve for ​y​ in terms of ​x​" really means is isolating ​y​ by itself on one side of the equation. In this case, divide both sides of the equation by 2. This gives you:

    y = 3x

    Next, check to see if there are any invalid ​x​ values. For example, if your equation involved a fraction such as 3/​x​, you'd use your knowledge that you can't have zero on the bottom of a fraction to tell you that ​x​ = 0 is not a member of the solution set.

    But with this example, ​y​ = 3​x​, there are no ​x​ values that would invalidate the equation. So you can choose any ​x​ values you want for the next part of the problem. For the sake of simplicity, use ​x​ = 1, 2, 3 for the next step.

    Substitute the ​x​ values from the last step into the equation, then solve to find each corresponding ​y​ value.

    \text{For } x = 1 \text{ you have } y = 3(1) \text{ or } y = 3 \\ \text{ For } x = 2 \text{ you have } y = 3(2) \text{ or } y = 6 \\ \text{ For } x = 3 \text{ you have } y = 3(3) \text{ or } y = 9

    So when given together, you have three sets of paired ​x​ and ​y​ values, or three points on a line:

    (1,3) (2,6) (3,9)

Graphing Your Solution Set

Now that you have your solution set, it's time to graph it. There's a little "algebra magic" involved here, because not every equation results in a straight line. But with the current example equation of ​y​ = 3​x​, you can use your knowledge of algebra to recognize that you're looking at the standard form for equation of a line

y = mx + b

where ​m​ = 3 and ​b​ = 0. So this equation does generate a straight line. That means you only need graph two points and connect them to define the line, although the third point is useful for checking your work.

Tips

  • Make sure you extend your line past the points you graphed. The usual notation is a small arrow at each end of the line, to show that it extends infinitely.

Graphing Inequalities as a Solution Set

The same process works for solving and graphing the solution set of an inequality. Consider that you're asked to solve and graph the inequality

-y ≥ 2x

You'll follow almost exactly the same steps as solving an equation, with a couple of quirks introduced by the presence of the inequality.

    To isolate ​y​ on its own, multiply (or divide) both sides by −1, which gives you:

    y ≤ -2x

    Tips

    • Look out – it's a trap! Did you remember that with inequality notation, multiplying or dividing both sides of the equation by a negative number means you have to flip the direction of the inequality sign?

    Using your knowledge of algebra, you can see that any value of ​x​ is possible. So while you could use any ​x​ values for the next step, it's convenient and simple to use ​x​ = 1, 2, 3 again.

    Solve for ​y​ values, using the ​x​ values you chose in the previous step.

    \text{ So, for } x = 1 \text{, you have }y ≤ -2(1) \text{ or } y ≤ -2 \\ \text{ For } x = 2 \text{, you have } y ≤ -2(2) \text{ or } y ≤ -4 \\ \text{ For } x = 3 \text{, you have } y ≤ -2(3) \text{ or } y ≤ -6

    Your paired solutions are:

    (1,-2) (2,-4) (3,-6)

    but don't forget about that ≤ inequality sign – it matters in the next step.

    First, graph the line depicted by the points in your solution set. Because your inequality sign ≤ reads as "less than or equal to," draw the line in solidly; it's part of your solution set. If you were dealing with the strict inequality <, which reads as "less than," you'd draw a dashed line because it isn't included in the solution set.

    Next, shade in everything underneath the slope of your line. Those are all the values "less than" the line, and your graph is complete.

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