um math prep: s8.1 - solving inequalities

Just as we can look at an equation to guess what numbers will make it true, we can solve some inequalities mentally by asking which numbers will make the inequality true.

Examples
Note how the translation of the equation in the left column leads to the solution in the right column.

to solve	we think	the solution
	"If x is larger than 1, the left-hand side is positive"
	"As long as x stays smaller than 6, the left-hand side will be less than 10"
	"When x gets larger than 3, the left-hand side will be negative"
	"If x=2 then x-2 is zero. When x is larger than 2, then x-2 is positive"
	"The value of x^2 is always positive, so x+5 needs to be positive"

Notation

When we want to describe intervals, we can sometimes use double inequalities: to describe all x values between a and b, we write a < x < b. (We assume a < b.) Note, however, that if we want to describe all values of x which are either less than a or greater than b we have to write it as two intequalities: x < a or x > b. Because a < b it is impossible to write a double inequality that describes both of x < a and x > b simultaneously.

Operations on inequalities

We solve inequalities using some of the techniques used in solving equations. However, there are some important differences in the application of these techniques to inequalities. In particular, we have to be careful that the operation we apply to both sides of the inequality leaves it unchanged.

Operations on inequalities
For a, b, and c real numbers: if a < b, then

		(we can add to either side and leave the inequality unchanged)
		(multiplying by a positive number leaves the inequality unchanged)
		(multiplying by a negative number reverses the inequality)
		(for positive numbers squaring both sides leaves the inequality unchanged)
		(for positive numbers, inverting both sides reverses the inequality)

Examples
Notice how the following examples are similar to and different from solving equalities.

Solve for x:

Using the rules above
Solve for x:

We can operate on all three sections of the inequality at once as long as we're careful:

Solving linear inequalities

Notice how in the previous examples we used the same techniques we use to solve linear equations: we isolate the variable on one side of the inequality. With inequalities, however, we have to be careful about reversing the inequality when multiplying or dividing by a negative number.

Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

Solve for x: