Lesson Explainer: Solving Systems of Linear Inequalities Mathematics

In this explainer, we will learn how to solve systems of linear inequalities by graphing them and identify the regions representing the solution.

A system of inequalities (represented by <,≀,>, and β‰₯) is a set of two or more linear inequalities in several variables and they are used when a problem requires a range of solutions and there is more than one constraint on those solutions.

In a graph representing a system of inequalities, shading above means greater than while shading below means less than. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

Inequalities of the form π‘₯>π‘Ž or π‘₯<π‘Ž will be represented as a vertical dashed line at π‘₯=π‘Ž (parallel to the 𝑦-axis) since the line itself is not included in the region representing the inequality, and the shaded region will be either on the right, for π‘₯>π‘Ž, or on the left, for π‘₯<π‘Ž. The same would apply for π‘₯β‰₯π‘Ž or π‘₯β‰€π‘Ž, except that now, the region would also include the line π‘₯=π‘Ž, which would be represented by a solid line, but the direction of shading would be the same.

Similarly, inequalities of the form 𝑦>𝑏 or 𝑦<𝑏 will be represented as a horizontal dashed line at 𝑦=𝑏 (parallel to the π‘₯-axis) since the line itself is not included in the region representing the inequality, and the shaded region will be either above, for 𝑦>𝑏, or below, for 𝑦<𝑏, the line 𝑦=𝑏. The same would apply for 𝑦β‰₯𝑏 or 𝑦≀𝑏, except that now, the region would also include the line 𝑦=𝑏, which would be represented by a solid line, but the direction of shading would be the same.

For example, consider the inequalities π‘₯β‰₯3 and 𝑦<5 represented on a graph:

The inequality π‘₯β‰₯3 is a solid line at π‘₯=3, since we have β‰₯; hence, the line itself is included in the region and the shaded region is on the right of the line, representing all values of π‘₯ greater than 3. If we had π‘₯>3, we would have the same thing, except that the line at π‘₯=3 would be dashed as it would not itself be included in the region. For π‘₯≀3 or π‘₯<3, the shading would be on the left, representing all numbers less than 3, and the line would be solid or dashed respectively, depending on whether the line π‘₯=3 is included in the region.

The inequality 𝑦<5 is represented as a dashed line at 𝑦=5, since we have <; hence, the line itself is not included in the region and the shaded region is below the line, representing all values of 𝑦 less than 5. If we had 𝑦≀5, we would have the same thing, except that the line at 𝑦=5 would be solid as it would itself be included in the region. For 𝑦β‰₯5 or 𝑦>5, the shading would be on the left, representing all numbers less than 5, and the line would be solid or dashed respectively, depending on whether the line 𝑦=5 is included in the region.

The intersection of the regions of each of the inequalities in a system is where the set of solutions lie, as this region satisfies every inequality in the system. We only include the values at the edges of intersections of the region if there is a solid line on both, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, excludes it from the solution set. For the example above, the two lines intersect at the point (3,5), but this is excluded from the solution set since it does not satisfy the strict inequality 𝑦<5.

The first quadrant can be represented by nonnegative values of π‘₯ and 𝑦 and, hence, the region where π‘₯β‰₯0 and 𝑦β‰₯0. Let’s consider an example, to see how this is visually interpreted from a graph.

Example 1: Determining the System of Inequalities Represented by a Given Graph

State the system of inequalities whose solution is represented by the following graph.

Answer

The shaded area represents all nonnegative values of π‘₯ and 𝑦, which can be translated to the inequalities π‘₯β‰₯0,𝑦β‰₯0.

In order to see this, let’s consider each inequality separately and see where they overlap. π‘₯β‰₯0, which is all nonnegative values of π‘₯ including the 𝑦-axis, is shaded in the first and fourth quadrants.

Similarly, 𝑦β‰₯0, which is all nonnegative values of 𝑦 including the π‘₯-axis, is shaded in the first and second quadrants.

The region where both inequalities overlap is in the first quadrant, represented by where the shaded regions of each inequality overlap.

The overlapping region is exactly the solution represented by the graph given.

We may have multiple inequalities of this form, bounding the values from above and/or below. For example, if we had the system of inequalities 2<π‘₯≀6,βˆ’2≀𝑦<7, where the second inequality is all the values of 𝑦 between βˆ’2 and 7, which can also be written seperately as 𝑦β‰₯βˆ’2 and 𝑦<7. This system of inequalities can be represented as follows:

Now, there is a solid line at 𝑦=βˆ’2 but a dashed line at 𝑦=7, which shows that 𝑦=βˆ’2 is included in the region, while 𝑦=7 is not, as shown in blue in the plot above. Again, the set of solutions for the system of inequalities is where the shaded regions of the inequalities intersect.

There are four points of intersection at (2,7), (2,βˆ’2), (6,7), and (6,βˆ’2) at the edge of the regions. However, only the point (6,βˆ’2) is included in the solution set, since the other points do not satisfy the strict inequalities.

We can also have inequalities with the equation of a line. For example, an inequality of the form 𝑦β‰₯π‘šπ‘₯+𝑐 is presented by a solid line, where the shaded region will be above the straight line 𝑦=π‘šπ‘₯+𝑐, whereas the inequality 𝑦>π‘šπ‘₯+𝑐 has the same shaded region but the boundary is presented by a dashed line. Similarly, the same would apply for π‘¦β‰€π‘šπ‘₯+𝑐 or 𝑦<π‘šπ‘₯+𝑐, except that the shaded region would be below the straight line. For example, the region for 2π‘₯+3𝑦>30, which is equivalent to 𝑦>βˆ’23π‘₯+10 in the form above, would be as follows:

Meanwhile, the region for 2π‘₯+3𝑦≀30 or π‘¦β‰€βˆ’23π‘₯+10 would be shaded below with a solid line. If there are multiple inequalities (i.e., a system of inequalities), then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities.

Let’s consider an example where we state the system of inequalities represented by a given graph.

Example 2: Determining the System of Inequalities Represented by a Given Graph

State the system of inequalities whose solution is represented by the following graph.

Answer

Recall that, in a graph representing a system of inequalities, shading above or to the right means greater than, while shading below or to the left means less than a particular line defined by π‘₯=π‘Ž, 𝑦=𝑏, or the general line 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line for both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

The shaded region is in the first quadrant for all nonnegative values of π‘₯ and 𝑦, which can be translated as the inequalities π‘₯β‰₯0,𝑦β‰₯0.

The vertical lines parallel to the 𝑦-axis are π‘₯=3 and π‘₯=6. Since the boundary on the left of the red region, at π‘₯=3, is represented by a solid line and the boundary on the right of the red region, at π‘₯=6, is represented by a dashed line, we have the inequalities π‘₯β‰₯3 and π‘₯<6, which is equivalent to 3≀π‘₯<6.

Similarly, the horizontal lines parallel to the π‘₯-axis are 𝑦=2 and 𝑦=6. Since the lines on both sides of the blue region are solid, we have the inequalities 𝑦β‰₯2 and 𝑦≀6, which is equivalent to 2≀𝑦≀6.

The equation of the line that passes through the points (8,0) and (0,8) is given by 𝑦=8βˆ’π‘₯, which is a solid line on the graph. Since the shaded region, in yellow, is below this line, we have the inequality 𝑦≀8βˆ’π‘₯, which can be rearranged as π‘₯+𝑦≀8.

Thus, the system of inequalities represented by the graph is π‘₯β‰₯0,𝑦β‰₯0,3≀π‘₯<6,2≀𝑦≀6,π‘₯+𝑦≀8.

Now, let’s consider another system of inequalities that includes the equation of a line. Consider the system of inequalities π‘₯>3,𝑦≀6,π‘₯+𝑦≀10.

The inequality π‘₯>3 is shown by a dashed line at π‘₯=3 and a shaded region (in red) on the right, and the inequality 𝑦≀6 is shown by a solid line at 𝑦=6 and a shaded region (in blue) below. Finally, the inequality π‘₯+𝑦≀10 is shown by a solid line with the equation 𝑦=10βˆ’π‘₯ and a shaded region below (in green).

The shaded regions where they all intersect are where all of the inequalities in the system are satisfied; all the solutions can be found in that region.

Let’s consider an example where we determine an inequality of this type from a given graph and the shaded region that represents the solution set.

Example 3: Determining the Inequality Represented by a Given Graph

The shaded area that represents the solution set of the inequalities 𝑦β‰₯3 and π‘₯β‰₯0 is .

  1. 2𝑦+π‘₯+8≀0
  2. 2𝑦+π‘₯βˆ’8≀0
  3. 𝑦+2π‘₯βˆ’8≀0
  4. 𝑦+2π‘₯+8≀8

Answer

The equation of the line that passes through (0,4) and (8,0) is given by 2𝑦=8βˆ’π‘₯. Since the shaded region lies below this line, this represents the region 𝑦≀4βˆ’12π‘₯, which is equivalent to the inequality 2𝑦+π‘₯βˆ’8≀0.

This is option B.

Now, let’s look at a few examples to practice and deepen our understanding to solve systems of linear inequalities by graphing them and identify the regions representing the solution.

The first few examples involve determining the system of inequalities from the region represented on a graph. The next example involves a region bounded by two straight lines.

Example 4: Determining the System of Inequalities Represented by a Given Graph

State the system of inequalities whose solution is represented by the following graph.

Answer

Recall that, in a graph representing a system of inequalities, shading above means greater than, while shading below means less than a general line defined by 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line on both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

The equation of the line that passes through the points (0,3) and (βˆ’1,0) is 𝑦=3π‘₯+3, which is a solid line on the graph. Since the shaded region, in yellow, is above this line, we have the inequality 𝑦β‰₯3π‘₯+3.

Similarly, the equation of the line that passes through the points (0,βˆ’8) and (βˆ’4,4) is 𝑦=βˆ’3π‘₯βˆ’8, which is a dashed line on the graph. Since the shaded region, in red, is above this line, we have the inequality 𝑦>βˆ’3π‘₯βˆ’8.

Thus, the system of inequalities represented in the graph is given by 𝑦β‰₯3π‘₯+3,𝑦>βˆ’3π‘₯βˆ’8.

In the next example, we will determine the system of inequalities that describes a region in a graph bounded by three straight lines.

Example 5: Writing a System of Inequalities That Describes a Region in a Graph

Find the system of inequalities that forms the triangle shown in the graph.

Answer

Recall that, in a graph representing a system of inequalities, shading above means greater than, while shading below means less than a general line defined by 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line on both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

The equation of the line that passes through the origin and intersects the other lines at (βˆ’2,2) and (2,βˆ’2) is 𝑦=βˆ’π‘₯, which is a solid line on the graph. Since the shaded region is above this line, we have the inequality 𝑦β‰₯βˆ’π‘₯.

Similarly, the equation of the line with a positive gradient that intersects the other lines at (1,8) and (βˆ’2,2) is 𝑦=2π‘₯+6, which is a dashed line on the graph. Since the shaded region is below this line, we have the inequality 𝑦<2π‘₯+6.

Finally, the equation of the line with a negative gradient that intersects the other lines at (1,8) and (2,βˆ’2) is 𝑦=βˆ’10π‘₯+18, which is a solid line on the graph. Since the shaded region is below this line, we have the inequality π‘¦β‰€βˆ’10π‘₯+18.

Thus, the system of inequalities represented by the graph is 𝑦<2π‘₯+6,𝑦β‰₯βˆ’π‘₯,π‘¦β‰€βˆ’10π‘₯+18.

Now, let’s look at a few examples where we identity particular regions shown on a graph from a given system of inequalities instead of determining them from the graph. In the next example, we will identify the region that represents the solution to a single inequality.

Example 6: Identifying Regions That Represent the Solutions to a System of Inequalities

Which of the regions on the graph contain the solutions to the inequality 𝑦β‰₯2π‘₯βˆ’4?

Answer

Recall that, in a graph representing a system of inequalities, shading above means greater than, while shading below means less than a general line defined by 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line on both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

In the graph, there are three distinct lines on the boundaries of the regions shown. Two of the lines are dashed, while one is solid. There are two lines with a positive gradient, one of which passes through the origin, and a third one with a negative gradient.

The inequality 𝑦β‰₯2π‘₯βˆ’4 can be represented by a solid line, since the boundary of the region, 𝑦=2π‘₯βˆ’4, is included in the region and the shaded area will be the region above the line due to the inequality β‰₯. This is the solid line that passes through the points (0,βˆ’4) and (2,0) with a positive gradient, as shown on the graph.

Thus, the regions on the graph that contain solutions to the inequality 𝑦β‰₯2π‘₯βˆ’4 are A, B, C, and D.

Now, let’s consider an example where we identify the regions that represent solutions to a system of inequalities, this time defined by two straight lines.

Example 7: Identifying Regions That Represent the Solutions to a System of Inequalities

Which regions on the graph contain solutions that satisfy both of the inequalities 𝑦<π‘₯,𝑦β‰₯2π‘₯βˆ’4?

Answer

Recall that, in a graph representing a system of inequalities, shading above means greater than, while shading below means less than a general line defined by 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line on both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

The inequality 𝑦<π‘₯ can be represented by a dashed line, since the boundary of the region, 𝑦=π‘₯, is not included in the region and the shaded area will be the region below the line due to the inequality <. This is the dashed line that passes through the origin with a positive gradient.

Similarly, the inequality 𝑦β‰₯2π‘₯βˆ’4 can be represented by a solid line, since the boundary of the region, 𝑦=2π‘₯βˆ’4, is included in the region and the shaded area will be the region above the line due to the inequality β‰₯. This is the solid line that passes through the points (0,βˆ’4) and (2,0), as shown on the graph.

Thus, the regions on the graph that contain solutions to the system of inequalities 𝑦<π‘₯ and 𝑦β‰₯2π‘₯βˆ’4 are C and D.

Finally, let’s consider an example where we identify the region that represents the solutions to a system of inequalities represented by three inequalities.

Example 8: Identifying Regions That Represent the Solutions to a System of Inequalities

Which region on the graph contains solutions to the set of inequalities 𝑦>2,𝑦β‰₯βˆ’π‘₯,π‘₯<1?

Answer

Recall that, in a graph representing a system of inequalities, shading above or to the right means greater than, while shading below or to the left means less than a particular line defined by π‘₯=π‘Ž, 𝑦=𝑏, or the general line 𝑦=π‘šπ‘₯+𝑏. In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.

If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system. We only include the edges of intersections of all the inequalities in the solution set if we have a solid line on both lines, as all inequalities need to be satisfied and a strict inequality, represented by a dashed line, on either or both sides would exclude it from the solution set.

The inequality 𝑦>2 can be represented by a dashed line, since the boundary of the region, 𝑦=2, is not included in the region and the shaded area will be the region above the line due to the inequality >. This is the dashed line parallel to the π‘₯-axis, as shown on the graph.

Similarly, the inequality 𝑦β‰₯βˆ’π‘₯ can be represented by a solid line, since the boundary of the region, 𝑦=βˆ’π‘₯, is included in the region and the shaded area will be the region above the line due to the inequality β‰₯. This is the solid line that passes through the origin with a negative gradient.

Finally, the inequality π‘₯<1 can be represented by a dashed line, since the boundary of the region, π‘₯=1, is not included in the region and the shaded area will be the region below the line due to the inequality <. This is the dashed line parallel to the 𝑦-axis, as shown on the graph.

The region that satisfies all of the inequalities will be the intersection of all the shaded regions of the individual inequalities.

Thus, the region on the graph that contain solutions to the system of inequalities 𝑦>2,𝑦β‰₯βˆ’π‘₯,π‘₯<1 is D.

Key Points

  • Shading to the right means greater than, while shading to the left means less than a particular line parallel to the 𝑦-axis defined by π‘₯=π‘Ž.
  • Shading above means greater than, while shading below means less than a particular line parallel to the π‘₯-axis defined by 𝑦=𝑏.
  • Shading above means greater than, while shading below means less than the general line defined by 𝑦=π‘šπ‘₯+𝑏.
  • The line itself is not included in the shaded region if we have a strict inequality.
  • If there is a system of inequalities, then the possible solutions will lie inside the intersection of the shaded regions for all the inequalities in the system.
  • In addition, we should also take the boundary of the region into account, where a solid line means equal to, while a dashed line means not equal to.
  • The intersection of the boundaries is included in the solution set only if both lines are solid (i.e., they contain no strict inequalities).

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