The average weight gain of an infant, after 6 months of age, is one pound a month, until the age of 2. If the average 6-month-old weighs 16 pounds, up to what age would an infant weigh 25 pounds or less?

### Basic Inequalities

Solving a linear inequality is very similar to solving a linear equality, or equation. There are a few very important differences. We no longer use an equal sign. There are four different inequality signs, shown below.

Notice that the line underneath the

Notice that the circle at -1 is *open*. This is to indicate that -1 is *not* included in the solution. A < sign would also have an open circle. If the inequality was a

Let's determine whether

Plug in -8 for

Of course, 2 cannot be greater than 3. Therefore, this is not a valid solution.

Now, let's solve the following basic inequalities.

- Solve and graph the solution to
2x−5≤17 .

For the most part, solving an inequality is the same as solving an equation. The major difference will be addressed in problem #2 below. This inequality can be solved just like an equation.

Test a solution,

Plotting the solution, we get:

Always test a solution that is in the solution range. It will help you determine if you solved the problem correctly.

- Solve and graph
−6x+7≤−29 .

When solving inequalities, be careful with negatives. Let’s solve this problem the way we normally would an equation.

Let’s check a solution. If

This is not a true inequality. To make this true, we must *flip* the inequality. Therefore, **whenever we multiply or divide by a negative number, we must flip the inequality sign.** The answer to this inequality is actually

This is true. The graph of the solution is:

### Examples

#### Example 1

Earlier, you were asked to find the last age (the maximum age) that an infant would weigh 25 pounds or less.

First, write an inequality. Let *m* represent the age of the infant, in months. Remember, when you get the final answer, you must add 6 for the initial weight of the infant at 6 months.

Adding 6, we have

#### Example 2

Is

Plug -5 into the inequality.

This is true because 22 is larger than 12. -5 is a solution.

#### Example 3

Solve and graph the solution to

No negatives with the

Test a solution,

The graph looks like:

#### Example 4

Solve and graph the solution to

In this inequality, we have a negative

Test a solution,

Notice that we *flipped* the inequality sign when we *divided* by -1. Also, this equation can also be written

Here is the graph:

### Review

Solve and graph each inequality.

x+5>−6 2x≥14 4<−x 3x−4≤8 21−8x<45 9>x−2 12x+5≥12 54≤−9x −7<8+56x 10−34x<−8 - \begin{align*}4x+15 \ge 47\end{align*}
- \begin{align*}0.6x-2.4<4.8\end{align*}
- \begin{align*}1.5>-2.7-0.3x\end{align*}
- \begin{align*}-11<12x+121\end{align*}
- \begin{align*}\frac{1}{2}-\frac{3}{4}x \le -\frac{5}{8}\end{align*}

For questions 16 and 17, write the inequality statement given by the graph below.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.10.