Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables, which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality [latex]x>3[/latex] is any number greater than 3. We show this on the number line below by shading in the number line to the right of 3 and putting an open parenthesis at 3.

Similarly, linear inequalities in two variables have many solutions. Any ordered pair [latex](x,y)[/latex] that makes an inequality true when we substitute in the values is a solution to a linear inequality.

Recognize the Relation between the Solutions of an Inequality and Its Graph

Now we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line shown previously again. The point [latex]x=3[/latex] separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3, all the numbers are greater than 3. See the figure below.

The solution to [latex]x>3[/latex] is the shaded part of the number line to the right of [latex]x=3[/latex].

Similarly, the line [latex]y=x+4[/latex] separates the plane into two regions.

On one side of the line are points with [latex]y \lt x+4[/latex].

On the other side of the line are the points with [latex]y > x+4[/latex].

We call the line [latex]y=x+4[/latex] a boundary line.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not the line is included in the solution.

[latex]\begin{array}{ll} Ax+By \lt C & Ax+By \leq C\\ Ax+By > C & Ax+By \geq C\\ \text{Boundary line is }Ax+By=C. & \text{Boundary line is }Ax+By=C.\\ \text{Boundary line is not included in solution.} & \text{Boundary line is included in solution.}\\ \text{Boundary line is dashed.} & \text{Boundary line is solid.}\\ \end{array}[/latex]

Now let’s take a look at what we found in the figure below. We’ll start by graphing the line [latex]y=x+4[/latex], and then we’ll plot the five points we tested, as shown in the graph.

We found that some of the points were solutions to the inequality [latex]y>x+4[/latex] and some were not.

Which of the points we plotted are solutions to the inequality [latex]y>x+4[/latex] ?

The points [latex](1,6)[/latex] and [latex](-8,12)[/latex] are solutions to the inequality [latex]y>x+4[/latex]. Notice that they are both on the same side of the boundary line [latex]y=x+4[/latex].

The two points [latex](0,0)[/latex] and [latex](-5,-15)[/latex] are on the other side of the boundary line [latex]y=x+4[/latex], and they are not solutions to the inequality [latex]y>x+4[/latex]. For those two points, [latex]y \lt x+4[/latex].

What about the point [latex](2,6)[/latex]? Because [latex]6=2+4[/latex] the point is a solution to the equation [latex]y=x+4[/latex] but not a solution to the inequality [latex]y>x+4[/latex]. So the point [latex](2,6)[/latex] is on the boundary line.

Any point you choose above the boundary line is a solution to the inequality [latex]y>x+4[/latex]. All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with [latex](0,0)[/latex] and [latex](-5,-15)[/latex], are not solutions to [latex]y>x+4[/latex] as shown in the figure.

The graph of the inequality [latex]y>x+4[/latex] is shown below.

The line [latex]y=x+4[/latex] divides the plane into two regions. The shaded side shows the solutions to the inequality [latex]y>x+4[/latex].

The points on the boundary line, [latex]y=x+4[/latex], are not solutions to the inequality [latex]y>x+4[/latex], so the line itself is not part of the solution. We show that by making the line dashed, not solid.

Example 2

Now graph the inequality [latex]y> 2x-1[/latex].

The solution and steps are given below.

The line [latex]y=2x-1[/latex] is the boundary line. On one side of the line are the points with [latex]y > 2x-1[/latex], and on the other side of the line are the points with [latex]y \lt 2x-1[/latex].

Let’s test the point [latex](0,0)[/latex] and see which inequality describes its position relative to the boundary line.

At [latex](0,0)[/latex], which inequality is true: [latex]y > 2x-1[/latex] or [latex]y \lt 2x-1[/latex]?

[latex]\begin{array}{rclcccrcl} y & > & 2x-1 & & & & y & \lt & 2x-1\\ 0 & \stackrel{?}{>} & 20-1 && & & 0 & \stackrel{?}{\lt} & 20-1\\ 0 & > & -1 & & & & 0 & \lt & -1\\ \end{array}[/latex]

Since [latex]y > 2x-1[/latex] is true, the side of the line with [latex](0,0)[/latex] is the solution. The shaded region shows the solution of the inequality [latex]y > 2x-1[/latex].

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality [latex]y > 2x-1[/latex].

We could use any point as a test point, provided it is not on the line. Why did we choose [latex](0,0)[/latex]? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that [latex]y \lt 2x-1[/latex].