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Checking Solutions to Inequalities

Substitute values for variables to check inequality solutions

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Solving Basic Inequalities

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.

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

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

\begin{align*} \le \end{align*} Less than or equal to

\begin{align*} \ge \end{align*} Greater than or equal to

Notice that the line underneath the \begin{align*}\le\end{align*} and \begin{align*}\ge\end{align*} signs indicates “equal to.” The inequality \begin{align*}x>-1\end{align*} would be read, “\begin{align*}x\end{align*} is greater than -1.” We can also graph these solutions on a number line. To graph an inequality on a number line, shading is used. This is because an inequality is a range of solutions, not just one specific number. To graph \begin{align*}x>-1\end{align*}, it would look like this:

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*} or \begin{align*}\le\end{align*} sign, then the circle would be closed, or filled in. Shading to the right of the circle shows that any number greater than -1 will be a solution to this inequality.

Let's determine whether \begin{align*}x=-8\end{align*} is a solution to \begin{align*}\frac{1}{2}x+6>3\end{align*}.

Plug in -8 for \begin{align*}x\end{align*} and test this solution.

\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.

Now, let's solve the following basic inequalities.

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

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.

\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.

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

When solving inequalities, 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*}

Let’s check a solution. If \begin{align*}x\end{align*} is less than or equal to 6, 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:

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.

\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.

Example 2

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

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.

Example 3

Solve and graph the solution to \begin{align*}\frac{3}{8}x+5<26\end{align*}.

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:

Example 4

Solve and graph the solution to \begin{align*}11<4-x\end{align*}.

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:

Review

Solve and graph each inequality.

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

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Vocabulary

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are <, >, \le, \ge and \ne.

solution

A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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