Inequalities are like equations only they do not have equals signs. In place of the equals sign is a less than, greater than, or a few other signs. Following are the signs encountered with inequalities.

 

< less than

> greater than

≤ less than or equal to

³ greater than or equal to

 

The inequality, x > 3, is read as “x is greater than 3.” This means that x can be 4, 5, 6, 6.5, 6.8, ….. or any number that is larger than 3.

 

If the inequality was x ≤ -2, it is read as “x is less than or equal to –2, this would include –2, -3, -3.7, -4,…. or any number that is less than or equal to –2.

 

Sometimes the inequality can be written in reverse order. By this it is meant that the variable is on the other (right) side. The inequality, 7 < x, can be read as “7 is less than x”. A better way to read this is by starting at the variable, x, and then reading towards the 7, thus reading the inequality backwards. Start at the x and moving left, the sign would then be reversed and would be greater than. Therefore, the inequality would be read as “x is greater than 7. Using that idea, the inequality –3 ³ x, could and should be read as “x is less than or equal to –3”.

 

When solving an inequality, treat it just like an equation. There is one special rule when solving inequalities that will be discussed later.

 

Example

 

x + 3 ³ 2

 

Subtract 3 from both sides.

x ³ -1

 

Answer! x is greater than or equal to –1.

 

 

 

 

Example

3x + 6 < 12 Subtract 6 from both sides.

3x < 6 Divide both sides by 3.

x < 2 Answer! x is less than 2.

 

The one special rule that was mentioned earlier is “when multiplying or dividing an inequality by a negative number, switch the sign”. Switching the sign means to change it from less than to greater than, and vice-versa.

 

 

 

Example

3 – 2x ³ -7

Subtract 3 from both sides.

 

– 2x ³ -10

 

Divide both sides by –2. Since –2 is negative, switch the sign.

 

x ³ 5

 

Answer! x is greater than 5.

 

 

 

 

 

Example

 

x + 2 < 3 x + 6

 

Subtract 3x from both sides to get all the x’s on the left. Also, subtract 2 from both sides.

 

-2x < 4

 

Divide both sides by –2, and switch the sign.

 

x < -2

 

Answer! x is less than –2.

 

 

 

 

Example

3 ( 2 x +1) > 7

 

Distribute the 3.

6 x + 3 > 7

 

Subtract 3 from both sides.

6 x > 4

 

Divide both sides by 6.

x >

 

Simplify.

 

x >

 

Answer!

 

Note- The sign was not switched in the previous problem because there was never division or multiplication of a negative number.