In Calculus, you will often need to work with inequalities. In this lesson you will review how to find solutions to inequalities.

First, let's review the rules for inequalities.

Recall that you can add any number to both sides of an inequality.

You can also add any two inequalities.

First, we subtract 2x from both sides of the inequality.

Then we subtract 9 from both sides.

Now, to isolate x, we need to divide both sides by -3. Because we are dividing by a negative number, we must change the direction of the inequality.

So, the solution set to this inequality is all real numbers less than 2.

You can also find the solution set for this inequality by graphing the two equations f(x) = 9 - x and g(x) = 3 + 2x.

From the graph, you can see that when x is less than two, the graph of f(x) is above the graph of g(x). Therefore, f(x) is greater than g(x) for values of x less than 2.

Find the solution set for the inequality shown. Write your answer using interval notation. Click "Submit" when finished.

Now, let's find the solution for a more complicated inequality. Let's see how you can solve the inequality x2 - 3x - 5 £ -1.

One way to solve this inequality is to graph the function f(x) = x2 - 3x - 5 and g(x) = -1 on the same axes and find the x-values for which the graph of f(x) is below the graph of g(x).

To do this, you need to solve the quadratic equation x2 - 3x - 5 = -1 to find the points of intersection of f(x) and g(x) .

Now that you have found the points of intersection of the two curves, you can see that f(x) g(x) when x is between -1 and 4.

Another way to solve this inequality is to use a sign chart. In this method, you need to rewrite the inequality so that all the nonzero terms are on one side of the inequality sign.

You know that the corresponding equation x2 - 3x - 4 = 0 has two solutions, 4 and -1.

The numbers -1 and 4 divide the real number line into three intervals. On each of these intervals, determine whether h(x) = x2 - 3x - 4 is positive or negative.

When x is less than -1, say, x = -2, h(x) is (-2)2 -3(-2) - 4, which is a positive number. So, h(x) is positive when x < -1.

When x is between -1 and 4, say x = 0, h(x) is (0)2 - 3(0) - 4 which is a negative number. So, h(x) is negative when -1 < x < 4.

When x is greater than 4, say x = 5, h(x) is (5)2 - 3(5) - 5, which is a positive number. So, h(x) is positive when x > 4.

We need to find the intervals on which x2 - 3x - 4 is zero or negative.

Looking at the chart, you can see that this is true when x is in [-1, 4].

Solve the inequality x2 - 5x -4. Write your answer using interval notation. Click "Submit" when finished.

You can multiply both sides of an inequality by a positive number.

However, if you multiply both sides of an inequality by a negative number, you must reverse the direction of the inequality.

If you take the reciprocal of both sides, you must reverse the direction of the inequality, provided that the quantities on both sides are positive.

Now let's see how you can solve inequalities. Consider the inequality 9 - x > 3 + 2x. We want to find all the values of x that satisfy this inequality.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education