The basic symbols for inequalities are:

< less than

Example: x < 7 means all numbers less than 7.

> greater than

Example: x > 4 means that x can be all numbers greater than 4.

≤ less than or equal to

Example: x ≤ ½ means that x can be ½ or any number less than ½

≥ greater than or equal to

Example: x ≥ 0 means that x can equal 0 or be any number greater than 0

It can be helpful to think of the symbol like a Pac-man mouth. Just like in the arcade game, you always want the most points so you need to eat the biggest numbers. I had one teacher who called it an “alligator” and used to draw funny teeth inside! However you choose to think about it, it’s important to remember that the “mouth” of the inequality symbol will always open towards the bigger number.

What makes inequalities different from equations is that inequalities represent a range of possible solutions. There will be multiple correct answers for x. The inequality you are left with when you simplify is simply an expression for the total range of x.

When you solve an inequality, you solve exactly the same way you do as an equation.

For example:

2x – 4 > 12
+ 4 + 4
2x > 16
/2 /2
x > 8

If this was an equation, our answer would have been x = 8. Here, x cannot equal 8 because there is no “equals to” symbol below the “mouth.” For this inequality, x can equal any value greater than 8, but not 8 itself.

If we were to graph this on a number line, we would draw an open circle at the number 8 and an arrow extending to the right. It must be an open circle to indicate that 8 itself is not a possible value. The arrow extends to the right because that is where the numbers greater than 8 are located. If the answer was x ≥ 8, we would draw a closed circle (a circle bubbled in like on the SAT answer grid).

There is one important rule to remember when solving inequalities:

If you multiply or divide by a negative number, flip the direction of the inequality.

For example:

- 4x + 8 ≤ 32
- 8 -8
- 4x ≤ 24
/-4 /-4
x ≥ -6

When we divided the -4 by both sides of the inequality, we had to flip the sign. On a number line, we would draw a closed circle at -6 and an arrow extending to the right. Keep this rule in your head and make sure to graph your answers on a number line when you practice inequalities!