# Linear Inequalities

## Definition of inequalities

Remember that the sign ‘$<$’ means “is less than,” or in other words “is to the left of (on a number line).” For example, the picture below shows that $-3 < 2$, because $-3$ is to the left of 2 on the number line.

Similarly, the sign ‘$>$’ means “is greater than” and “is to the right of (on a number line).” So the same picture also shows that $2 > -3$.

Each row of the table to the right has a mathematical sentence in it. Use the menu in each row to choose the mathematical sign which makes that sentence true.

Also remember that the symbol ‘$≤$’ means “is less than or equal to” and the symbol ‘$≥$’ means “is greater than or equal to.” For example, $1 ≤ 2$, $1 ≥ -2$, and $1 ≤ 1$ are all true statements.

Each row of the table below compares two numbers. Indicate whether each statement involving those two numbers is true.

A mathematical sentence that uses one of these symbols ($<$, $>$, $≤$, or $≥$) is called an inequality.

## Graphing inequalities

In the rest of this lesson, you will study inequalities that use variables, like $y<1$ and $3x+2y<5$. Given such an inequality, we want to find all its solutions, which is called its solution set. However, you won’t be able to list all of the solutions. One way to describe all possible solutions is to graph them.

 All of the blue points on the grid to the left satisfy the equation $y=1$. Name a new point that satisfies $y=1$ (that is, a point with $y$-coordinate equal to 1).
 We want to graph the solution set of $y=1$. This means that we want to show all of the points whose $y$-coordinate is 1. Click to graph the solution set of $y=1$. What is the shape of the solution set (triangle, circle, line, square, …)?

As you saw in equalQuestion, the graph of the solution set of $y=1$ is a straight line. Now, let’s see what the solution set of $y<1$ looks like.

 The blue line on the grid is the graph of the equation $y=1$. All of the blue points on the grid have $y$-coordinate less than 1. Are these points above, below, or on the line $y=1$?
 All of the blue points on the grid are in the solution set of $y<1$. Name a new point that satisfies $y<1$ (that is, a point with $y$-coordinate less than 1).
 Is it possible to graph the solution set of $y<1$ using a straight line?
 Think about the graph of $y<1$. Do you think it forms a curve (like a circle), an entire region (like a filled-in box), or some other shape?

The convention we use to graph $y<1$ is to shade the region where all of the points in the solution set lie. So, the region below the line $y=1$ is shaded.

 Click to look at the graph of $y<1$. Does the point you found above lie in the shaded region?
 Now think about what the graph of the solution set of $y>1$ looks like. Does it form a curve, a region, or some other shape?
 Is the graph of $y>1$ above, below, or on the line $y=1$?

Here is the graph of $y<mx+b$. Leave $m$ at 0 and use the slider for $b$ to change its value. Notice how changing $b$ affects the solution set of $y<b$.

 Using the slider for $b$, find an inequality with $m=0$ that has all of the red points on the grid in its solution set. (The inequality is written below the grid.)
 Using the slider for $b$, find an inequality with $m=0$ that has none of the red points on the grid in its solution set.
 Slide $b$ to 0. Which red points are in the solution set of the inequality ($y<0$) now? , , , and

Now, leave $b$ set to 0. Use the slider for $m$ to change its value, and notice how changing $m$ affects the solution set of the inequality $y<mx$.

 Set $m$ to 1. What is the inequality you are looking at now?
 Which of the red points are in the solution set of this inequality? , , , , and
 Name the red points that aren’t in the solution set of this inequality. and

Now we want to see what the solution set of $x>1$ looks like.

 The line shown to the right of the $y$-axis is the graph of the equation $x=1$. All of the blue points on the grid have $x$-coordinate greater than 1. Are these points to the right of, to the left of, or on the line $x=1$?
 All of the blue points on the grid are in the solution set of $x>1$. Name a new point that satisfies $x>1$ (that is, a point with $x$-coordinate greater than 1).
 Is it possible to graph the solution set of $x>1$ using a straight line?
 Do you think the graph of $x>1$ forms a curve, an entire region, or some other shape?

The convention we use to graph $x>1$ is to shade the region where all of the points in the solution set lie. So, the region to the right of the line $x=1$ is shaded.

 Click to look at the graph of $x>1$. Does the point you found above lie in the shaded region?
 Do you think the graph of $x<1$ forms a curve, an entire region, or some other shape?
 If you graphed $x<1$, would it be to the right of, to the left of, or on the line $x=1$?

We want to find all of the points that satisfy the inequality $3x+2y<5$. The graph of $3x+2y=5$ is shown on the grid to the left. The solution set of $3x+2y<5$ will be the points on one side or the other of this line. Let’s consider the point $(0,0)$ and check if its coordinates satisfy $3x+2y<5$.

Substituting $(x,y) = (0,0)$ into $3x+2y<5$, we have:

$$\cl"tight"{\table 3(0),+,2(0),<, 5, \;?; 0,+,0,<, 5, \;?; \colspan 3 0, <, 5, \;?}$$

Is $0<5$? Yes. So the point $(0,0)$ is in the solution set.

Use the method shown above to check whether the point $(0,4)$ is in the solution set of $3x+2y<5$.

 Click to look at the graph of $3x+2y<5$. Is the point $(0,0)$ in the shaded region?
 Is the point $(0,4)$ in the shaded region?

So far, you have only looked at inequalities that involved the $>$ and $<$ signs. Now we will look at inequalities that involve the $≥$ and $≤$ signs.

Look back at the graph of $3x+2y<5$. Let’s see if the point $(3,-2)$ satisfies this inequality:

$$\cl"tight"{\table 3(3),+,2(-2),<,5,\;?; 9,+,(-4),<,5,\;?; ,5,,<,5,\;?,\;\;\text"No."}$$

Check whether the point $(1,1)$ satisfies the inequality $3x+2y < 5$.

Now, let’s look at the related inequality $3x+2y≤5$. Does the point $(-1,4)$ satisfy that inequality?

Click to see the graph of the solution set of $3x+2y≤5$. Find the points $(3,-2)$, $(1,1)$ and $(-1,4)$ on the graph.

 Is $(1,1)$ in the solution set of $3x+2y≤5$?

Notice that we graphed the solution set of $3x+2y≤5$ by including a solid line (the line $3x+2y=5$).

When an inequality involves the $≥$ or $≤$ signs, we graph its solution set by including a solid line at its boundary. This is because the points on the line are also in the solution set.

 The bottom grid to the left shows the solution set of the inequality $y ≥ mx+b$. Use the sliders for $m$ and $b$ below that grid to find the solution set shown on the top grid. What is the inequality with this solution set?