How to Solve and Graph Systems of Inequalities

I recently sent this as advice to one of my clients having trouble with linear systems of inequalities. I thought I would share it here on my blog for students, parents, and tutors who have use for it.


A system with regular lines (the ones with equals signs in them that you have done before) shows the single point where the two lines cross each other on the graph. The X and Y at that point are the two numbers that make the equal sign true. For instance, with the equations 3 = 5X +Y and 10 = 2X -Y, the answer is x = 7/13 and y = 4/13 because if you plug those numbers into both equations you get true statements, 3=3 and 10=10. The point (7/13, 4/13) is the point where the two lines cross each other. Inequalities, where you have "less than" or "greater than" signs work the same way. But, instead of getting a point where the equations are true, you get a whole area on the graph where they are true. So, the answer to a system of inequalities is a big shaded area on a graph, not just one point.


1. Put both equations from the problem into slope-intercept form (Y > mX + b or Y-3Y + 9, you would subtract 9 from both sides to make it 6X - 9 >-3Y. Then, you have to divide both sides by -3 and flip the sign because you are dividing by a negative number. This gives you Y < -2X -9. Make sure the Xs are in front and the plain number (called the constant) comes after. If you have Y < -9 -2X, you have to switch the order of the -9 and the -2X.

2. Pretend the "less than" and "greater than" signs are equal signs and graph the first equation on the coordinate plane as though it is a normal line with an equal sign. For instance, pretend that Y < -2X -9 is really Y =-2X -9. Find the "b," which is plain the number on the end. Here, b is -9. That is the point where the graph crosses the Y-axis (the up and down line). Draw a point at this spot. Then, find the number in front of X - this is the "m," or slope. It represents the steepness (rise over run) of the graph. If the "m" is a whole number, put it over one to turn it into a fraction. Slope is one of the few places in algebra where fractions actually make things easier! In my example, the slope is -2, so you would change it to -2/1. Put your pencil on the "b" you drew on your graph. Look at the top number of the slope fraction and move that many spaces up or down (move down if it is a negative number). Then, move forward the number of spaces represented by the bottom of the fraction. In the example, since the slope is -2/1, you would move 2 spaces DOWN and 1 space RIGHT. Draw your second point here.

3. Now, you can draw a line going through your two points and graph the equation.
Stop pretending that the sign in the equation was an equal sign and look to see whether the inequality sign (< or >) is opening towards the Y or away from the Y and whether it has an equal sign in it or not. If the sign is "less than or equal to" or "greater than or equal two" draw a solid line through the two points. But, if is just "less than" or "greater than" draw a dotted line. The dotted line says that the line itself is not part of the answer. It is just like when you use an open circle instead of a closed circle on the number line. If the sign is opening towards the Y, shade above the line. If it is opening away from the Y, shade below the line.

4. Do steps 2 and 3 for the second equation on the same graph. Now, you will have two parts of the graph shaded. Look to see where they overlap and shade this area extra dark because this is the answer, and you want it to be clear to the teacher.

EXTRA: Sometimes, the two areas won't overlap. If they don't over lap the answer is "no solution." Sometimes both areas are exactly the same. In this case the answer is "all real numbers" because any possible numbers for X in Y in one equation will also make the other equation true. If one of your equations is something like Y<7 or X>4, these are vertical line or horizontal lines. You graph them the way you would graph Y=7 (a horizontal line) or X =4 (a vertical line) and then shade over or under.


Great blog Kathryn! Just to share my 2 cents - The easiest method to graph a linear equation (straight line) is to find the X-intercept and the Y-intercept and then simply joining those two points. The orientation of the line will reveal the nature of the slope anyway (but I won't bother employing slope to find a point - although the concept is REALLY cool!!). Finding X-intercept and Y-intercept is possibly the easiest thing: Plug Y = 0 and solve for X for former and plug X=0 and solve for Y for the latter! Cheers and keep up the great work! I recently joined WyZant and and loving the experience so far :-)
can you help me how solve this Y=-1/2X-5 and graph too? I am a mother that i try understand.
You cannot really solve it. You will simply need to find any 2 points on this line and then you can join them to draw / graph the line. To find the first point (x intercept) put Y = 0 and solve for X. I can see that will be X = - 10
and then put x = 0 (Y intercept) and solve for Y, which will give Y = -5. As such, the two points are (-10,0) and (0,-5). Simply locate those 2 co-ordinates and then join them. You done!
X-intercept is called x-intercept as it is an intercept on X axis and has Y = 0.
Y-intercept is called y-intercept as it is an intercept on Y axis and has X = 0.
if (isMyPost) { }