Inequalities are mathematical expressions that show the relationship between quantities that are not equal. They're an important concept on the ACT Math® Section, often appearing in various question types. Let's look at how they work and how you'll see them on the Test.
Some Examples
Inequalities start out pretty basic, then get seemingly, but not actually, more difficult. Most students know the basics of inequalities.
If you're having trouble remembering which sign is less than vs greater than or remembering how to say either, remember this trick. It works for most students. The "less than" sign looks like an L <. The greater than sign > does not look like an L. The one that looks like an L means and is pronounced less than. The one that doesn't is greater than.
Inequalities get a little trickier with the negative signs – fret not. All you need to do is flip the inequality sign when you multiply or divide by a negative.
Click for the Answer
Subtract 4 from both sides:
-4 < -5x
Divide by -5 and flip the inequality sign:
> x
Simplify:
> x
The ACT Math® section will likely test you on this. Keep in mind that multiplying or dividing by a negative means flipping the inequality sign.
Graph of Inequalities
Now, let's tackle something that often trips students up on the ACT Math® section: graphing inequalities. This can confuse students because it's not the sort of thing that's taught regularly in most high schools. Like most questions on the ACT Math® section, graphs of inequalities are actually quite intuitive.
What would it look like if we took this expression and replaced the '=' with a greater than or less than sign?
It's the "same" line. We just changed the line to a dotted line and shade above because it's greater than. The expression to the right of the greater than sign should be literally greater than our y. Hence why we shade above the line, but not on the line. Our expression is not equal to, but greater than y.
For strict inequalities (> or <), we use a dotted line to show that the line itself is not included in the solution. For non-strict inequalities (≥ or ≤), we use a solid line to indicate that the line is part of the solution.
What about less than or equal to? It's the same principle.
Inequality 1:
Inequality 2:
Notice in the graph above, the line is solid - not dashed - because the sign is less than OR equal to - not less than.
How will this show up on the ACT Math® section?
The ACT Math® is going to ask you this as follows. Sometimes they just want to know which of the graphs below represents some equation with a greater than or less than sign, like we graphed above.
Sometimes they'll give you a line and an inequality. They may want to know which graphs represent the line and the inequality. That's the way the ACT Math® section almost always asks about graphing inequalities.
Understanding inequalities and their graphs is important for your success on the ACT Math® section. Practice identifying and graphing different types of inequalities to build your confidence with these questions.