Linear Inequalities and Graphs

This section focuses on properties on linear inequalities and how they are graphed.

# Linear Inequalities

A linear inequality is similar to the linear function, except in an inequality, the expressions are not equal to each other. Rather, in inequalities, the expressions are related either by one being greater than the other, less than the other, or the previous but including equal to. Perhaps these symbols look familiar: < > ≤ ≥ These symbols relate expressions to each other according to their value. Consider the inequality 3>1. This statement is read as three is greater than one. Here is another true statement: 3<5. Three is less than five. The other two symbols represent less than or equal to and greater than or equal to. This statement is not true: 3>3 because three is certainly not greater than itself. This is true, though: 3≥3. Three is greater than or equal to three. Whether the relation is equal to or not will affect the answer.

In a linear inequality, the x degree is also one. Let's consider this particular inequality of two variables: y>x+2. This literally means y is greater than two more than x. What ever x+2 is, y is capable of being greater than that value, but it can't be equal to or less than it. For instance, let's say that x is 3. y>(3)+2, which simplified is y>5. The variable y can be anything that's greater than five. So, for this example, y could be 6, 7, 5.5, 8, 12, 35, 1563, or even 5.001. However, y cannot be 3, 2, 0, -3, -5, -12, or even 5. So, y could be so many numbers! How is that represented? Well, graphing the inequality function is one good, solid way. Let us observe the properties of these graphs.

This is the graph of the inequality y>1/3x+4/3. Just like in linear functions, the 4/3 is the y-intercept, and the 1/3 is the slope. Notice two main differences between a function line and inequality line: This line is dashed, and the region above the line is shaded. How did this happen? Read the equation verbally. y is greater than one third x plus four thirds. y is greater than means that for any value of x after undergoing its necessary calculations, y must be more than it. So, any point in the shaded region satisfies the inequality. That is why the area above is shaded. The line is dotted because any value on the line when substituted as x and y becomes untrue in the inequality. If you took the point (-1,1) into the inequality, you get 1>1, which is not true. Some inequalities, such as 2x-3y≤6, have a solid line instead of a dashed one. The reason is this time, a point on the line substituted into the inequality makes it true. Now, we will discuss the components of a linear inequality in depth.

## Slope and Intercepts

The slope and intercepts of a linear inequality use the same rules of linear functions. The slope and intercept refer to the line part of the graph, not the shaded region. If the shaded region was considered, the inequality would not have any slope and infinite intercepts. Just like before, the slope is the rate of change in the line, and the intercepts are where the line part crosses the x and y-axes. To find the slope, you would simply use the slope formula along with two points that are on the line and not in the shaded region. If you use points in the shaded area, then your slope will be messed up. Only use points on the line! The intercepts undergo the same concept. The y-intercept is where the line crosses the y-axis, and the x-intercept is where the line crosses the x-axes. For more information on these topics, refer to Linear Functions and Graphs.