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?

### Watch This

First watch this video.

Khan Academy: One-Step Inequalities

Then watch this video.

Khan Academy: One-Step Inequalities 2

### Guidance

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

\begin{align*} < \end{align*}

\begin{align*} > \end{align*}

\begin{align*} \le \end{align*}

\begin{align*} \ge \end{align*}

Note that the line underneath the \begin{align*}\le\end{align*}

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 \begin{align*}\ge\end{align*}

#### Example A

Is \begin{align*}x=-8\end{align*}

**Solution:** Plug in -8 for \begin{align*}x\end{align*}

\begin{align*}\frac{1}{2}(-8)+6 &> 3\\ -4+6 &> 3\\ 2 &> 3\end{align*}

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

#### Example B

Solve and graph the solution to \begin{align*}2x-5 \le 17\end{align*}

**Solution:** For the most part, solving an inequality involves the same process as solving an equation.

\begin{align*}& 2x-\bcancel{5} \le 17\\ & \underline{\quad \ + \bcancel{5} \ +5 \; \;}\\ & \quad \ \frac{\bcancel{2}x}{\bcancel{2}} \le \frac{22}{2}\\ & \qquad x \le 2\end{align*}

Test a solution, \begin{align*}x = 0: 2(0)-5 \le 17 \rightarrow -5 \le 17 \checkmark\end{align*}

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.

#### Example C

Solve and graph \begin{align*}-6x+7 \le - 29\end{align*}.

**Solution:** This example involves the primary difference between solving equations and solving inequalities. When solving inequalities, you need to be careful with negatives.

Let’s solve this problem the way we normally would an equation.

\begin{align*}& -6x+\bcancel{7} \le -29\\ & \underline{\qquad \ \ -\bcancel{7} \quad \ -7 \; \; \; \;}\\ & \qquad \frac{-\bcancel{6}x}{-\bcancel{6}x} \le \frac{-36}{-6}\\ & \qquad \quad \ x \le 6\end{align*}

Now check a solution. Since \begin{align*}x\end{align*} should be less than or equal to 6 to make the statement true, let’s test 1.

\begin{align*}-6(1)+7 & \le -29\\ -6+7 & \le -29\\ 1 & \bcancel{\le} -29\end{align*}

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 \begin{align*}x\ge 6\end{align*}. Now, let’s test a number in this range.

\begin{align*}-6(10)+7 & \le - 29\\ -60+7 & \le -29\\ -60 & \le -29\end{align*}

This is true. The graph of the solution is:

**Intro Problem Revisit** 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.

\begin{align*}16 + m \le 25 \\ m \le 9 \end{align*}

Adding 6, we have \begin{align*}m \le 15\end{align*}. So, up to 15 months, the average baby should weigh 25 pounds or less.

### Guided Practice

1. Is \begin{align*}x = -5\end{align*} a solution to \begin{align*}-3x+7>12\end{align*}?

Solve the following inequalities and graph.

2. \begin{align*}\frac{3}{8}x+5<26\end{align*}

3. \begin{align*}11<4-x\end{align*}

#### Answers

1. Plug -5 into the inequality.

\begin{align*}-3(-5)+7 &>12\\ 15+7&>12\end{align*}

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

2. No negatives with the \begin{align*}x-\end{align*}term, so we can solve this inequality like an equation.

\begin{align*}& \frac{3}{8}x+\bcancel{5}<26\\ & \underline{\quad \ -\bcancel{5} \ \ -5 \; \; \; \; \; \;}\\ & \ \xcancel{\frac{8}{3} \cdot \frac{3}{8}}x<21 \cdot \frac{8}{3}\\ & \qquad \ x<56\end{align*}

Test a solution, \begin{align*}x = 16: \frac{3}{8}(16)+5 < 26 \checkmark\end{align*}

\begin{align*}6+5 < 26\end{align*}

The graph looks like:

3. In this inequality, we have a negative \begin{align*}x-\end{align*}term. Therefore, we will need to flip the inequality.

\begin{align*}& \ \ 11<\bcancel{4}-x\\ & \underline{-4 \ \ - \bcancel{4} \; \; \; \; \; \; \;}\\ & \frac{7}{-1} < \frac{\bcancel{-}x}{\bcancel{-1}}\\ & -7>x\end{align*}

Test a solution, \begin{align*}x = -10: 11 < 4-(-10) \checkmark\end{align*}

\begin{align*}11 < 14\end{align*}

Notice that we **FLIPPED** the inequality sign when we *DIVIDED* by -1. Also, this equation can also be written \begin{align*}x< -7\end{align*}. Here is the graph:

### Explore More

Solve and graph each inequality.

- \begin{align*}x+5>-6\end{align*}
- \begin{align*}2x \ge 14\end{align*}
- \begin{align*}4<-x\end{align*}
- \begin{align*}3x-4 \le 8\end{align*}
- \begin{align*}21-8x<45\end{align*}
- \begin{align*}9>x-2\end{align*}
- \begin{align*}\frac{1}{2}x+5 \ge 12\end{align*}
- \begin{align*}54 \le -9x\end{align*}
- \begin{align*}-7<8+\frac{5}{6}x\end{align*}
- \begin{align*}10-\frac{3}{4}x<-8\end{align*}
- \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.