How do you check if a point satisfies a system of linear inequalities?

1 Graphing a system

One way to check if a point satisfies a system of linear inequalities is to graph the system and see if the point lies in the shaded region. To graph a system, you need to plot the boundary lines of each inequality and shade the appropriate side of each line. For example, to graph y < 2x + 3, you would draw a dashed line for y = 2x + 3 and shade the region below the line. To graph y > x - 2, you would draw a solid line for y = x - 2 and shade the region above the line. The intersection of the shaded regions is the solution set of the system.

2 Substituting coordinates

Another way to check if a point satisfies a system of linear inequalities is to substitute the coordinates of the point into each inequality and see if the result is true. For example, to check if the point (1, 2) satisfies the system y < 2x + 3 and y > x - 2, you would replace x with 1 and y with 2 in both inequalities and simplify. You would get 2 < 5 and 2 > -1, which are both true, so the point is a solution. If any of the inequalities is false, then the point is not a solution.

3 Using a table

A third way to check if a point satisfies a system of linear inequalities is to use a table to compare the values of the linear functions at the point. For example, to check if the point (1, 2) satisfies the system y < 2x + 3 and y > x - 2, you would create a table with three columns: x, y, and the difference between y and each function. You would fill in the table with the coordinates of the point and the values of the functions at that point. You would then look at the signs of the differences and see if they match the inequalities. For example, if y < 2x + 3, then the difference should be negative. If y > x - 2, then the difference should be positive.

| x | y | y - (2x + 3) | y - (x - 2) |

|---|---|---------------|-------------|

| 1 | 2 | -2 | 1 |

Since the signs of the differences match the inequalities, the point is a solution.

4 Tips and tricks

To check if a point satisfies a system of linear inequalities, it is helpful to use parentheses when substituting coordinates to avoid errors with negative signs or operations. Additionally, a calculator or a spreadsheet can be used to perform calculations and compare values quickly and accurately. Using different colors or symbols to mark the shaded regions and the points on the graph can also help prevent confusion. Lastly, you can check your work by trying another point or method to confirm your answer.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?