How do you graph a linear inequality

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You can graph a linear or quadratic inequality similarly to how you would graph an equation. The difference is that, since an inequality shows a set of values greater than or less than, your graph will show more than just a dot on a number line or a line on a coordinate plane. By using algebra and evaluating the inequality sign, you can determine which values are included in the solution of an inequality.

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   Solve for the variable. To solve the inequality isolate the variable using the same algebraic methods you would use to solve an equation.[1] Remember that when you multiply or divide by a negative number, you need to flip the inequality sign.

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   Draw a number line. Include the relative value on your number line (the value you found the variable to be less than, greater than, or equal to). Make the number line as long or short as required.

   • For example, if you found that , make sure to include a point for 1 on the number line.

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   Draw an arrow indicating the included values. If the variable is greater than the relative value, your arrow should point to the right, since the solution includes values greater than that number. If the variable is less than the relative value, your arrow should point to the left, since the solution includes values less than that number.[3]

   • For example, for the solution , you would draw an arrow pointing to the right, since the solution includes values greater than 1.

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   Graph the line on a coordinate plane. To do this, turn the inequality into an equation, and graph as you would any equation of a line.[5] Plot the y-intercept, then use the slope to graph other points on the line.

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   Graph the line on a coordinate plane. To do this, turn the inequality into an equation, and graph the line as you normally would. Since you have a quadratic equation, the line will be a parabola.[9]

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   Find some test points. In order to determine which area to shade, you need to pick points from inside the parabola, and from outside the parabola.

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   Shade the appropriate area. To determine which area to shade, plug the values of and from your test points into the original inequality. Whichever point produces a true inequality indicates which area of the graph should be shaded in.[10]

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• Question

  How do I graph the inequality y is less than or equal to X + 6?

  

  Since you do not have an x-squared term, you are working with a linear inequality. Turn the inequality into the equation y = x + 6. Plot the line using this equation. Since it is in the form of y = mx + b, you can tell that the y-intercept is at the point (0, 6), and that the slope is 1, or 1/1. Since it is less than OR equal to, the line should be solid. Since y is less than the line, you would shade the area below the line.

• Question

  How do I graph x - y is greater than 5 as a linear inequality?

  

  First, substitute "greater than" with "equals," which gives you x - y = 5. Then rearrange the equation to get y = x - 5. Input some values for x. If you do 1, 5, 10, you get y-values -4, 0, 5. Plot these 3 points (1,-4), (5,0) and (10,5). Then pick a point on your graph (not on the line) and put this into your starting equation. Take the point (4,2) for example. You have 4 - 2 > 5, and 2 > 5 is false. This means that the graph area on the same side of the line as point (4,2) is not in the region x - y > 5. Therefore, any point on the other side of the line should be in this region. To check, use point (8,2). 8 - 2 > 5, 6 > 5, which is true. You can then draw arrows off your line indicating the region represented by x - y > 5.

• Question

  How do I graph y=5x+1?

  

  The Y-axis goes up and down; the X-axis goes left and right. Plug in x values: In the equation above, if x is 0, what is y? Plugging 0 in for x and doing the calculation gives a 1 for y. The first point, then, is (0,1). (When x is 0, y is 1.) Next, plug in 1 for x. (5 times 1) + 1 = 6. So the next point is (1,6). (When x is 1, y is 6.) Now, plug in 2 for x. (5 times 2) + 1 = 11. The next point is (2,11). Plot the points on the graph and draw a line through all of them.

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• Always simplify the inequalities before graphing them.

• If you get really stuck, you can input the inequality into a graphing calculator and try to work backwards.

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Article SummaryX

To graph linear inequalities, isolate the variable and solve the problem. Then, draw a vertical or horizontal number line that includes the value that the variable is greater or less than, and draw a circle on the tick mark of the relative value. Draw an arrow from that value to the greater or less than side of the number line, depending on the equation. For example, for an equation that solves as y > 1, you would draw a circle on 1 and an arrow pointing up, since the solution includes all values greater than 1. If you want to learn how to graph the inequality on a coordinate plane, keep reading the article!

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