then reload this page.
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 $2 > -3$.
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
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.
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.
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 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 $y=1$
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
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
Now we want to see what the solution set of $x>1$ looks like.
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.
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:
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 $3x+2y<5$.
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:
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.
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.