Inequalities on a number line

More than one answer

An equation like $x + 3 = 7$ has one solution. An inequality compares two expressions that might not be equal:

Hunt for solutions

Which numbers make $x > 3$ true? Tap numbers on the line to test them. Try some on each side of $3$, try $3$ itself, and try the halfway points like $3.5$.

$x > 3$

Tap a number on the line to test it.

Test every pointShow the solution set

Every number to the right of $3$ works: $3.5$, $4$, $10$, and even $3.001$. There are infinitely many, far too many to list. So instead of listing solutions, we draw them by shading the line.

$3$ itself is the interesting case. $3 > 3$ is false, so $3$ is not a solution, even though everything just past it is. The graph needs a way to show that.

Open and closed dots

Now test $x \geq 3$. What changes?

$x \geq 3$

Tap a number on the line to test it.

Test every pointShow the solution set

Only one point changed: $3$ itself is now a solution. The dot at the endpoint shows whether it's included:

• Open dot for $<$ and $>$: the endpoint is not a solution.
• Closed dot for $\leq$ and $\geq$: the endpoint is a solution.

Reading a graph

To go the other way, look at two things: which way it's shaded (left means less than, right means greater than) and whether the dot is filled in.

Shaded left, open dot: this is $x < 2$.

Shaded left, closed dot: this is $x \leq -1$.

The variable on the right

$5 > x$ means the same as $x < 5$. Read it as "$x$ is less than $5$." Flipping the whole statement around, sides and symbol together, doesn't change its meaning.

Key ideas

• An inequality usually has infinitely many solutions, so we graph them instead of listing them.
• $<$ and $>$ leave out the endpoint: use an open dot. $\leq$ and $\geq$ include it: use a closed dot.
• Shade left for less than and right for greater than (with $x$ on the left).