<meta http-equiv="refresh" content="1; url=/nojavascript/"> Checking Solutions to Inequalities ( Read ) | Algebra | CK-12 Foundation
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Checking Solutions to Inequalities

%
Best Score
Practice Checking Solutions to Inequalities
Practice
Best Score
%
Practice Now
Solving Basic Inequalities
 0  0  0

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. We no longer use an equal sign. There are four different inequality signs, shown below.

 < Less than

 > Greater than

 \le Less than or equal to

 \ge Greater than or equal to

Notice that the line underneath the \le and \ge signs indicates “equal to.” The inequality x>-1 would be read, “ x 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 x>-1 , 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 \ge or \le 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.

Example A

Is x=-8 a solution to \frac{1}{2}x+6>3 ?

Solution: Plug in -8 for x and test this solution.

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

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

Example B

Solve and graph the solution to 2x-5 \le 17 .

Solution: For the most part, solving an inequality is the same as solving an equation. The major difference will be addressed in Example C. This inequality can be solved just like an equation.

& 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

Test a solution, x = 0: 2(0)-5 \le 17 \rightarrow -5 \le 17 \checkmark

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 -6x+7 \le - 29 .

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

& -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

Let’s check a solution. If x is less than or equal to 6, let’s test 1.

-6(1)+7 & \le -29\\-6+7 & \le -29\\1 & \bcancel{\le} -29

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 x\ge 6 . Now, let’s test a number in this range.

-6(10)+7 & \le - 29\\-60+7 & \le -29\\-60 & \le -29

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 inital weight of the infant at 6 months.

16 + m \le 25 \\m \le 9

Adding 6, we have m \le 15 . So, up to 15 months, the average baby should weigh 25 pounds or less.

Guided Practice

1. Is x = -5 a solution to -3x+7>12 ?

Solve the following inequalities and graph.

2. \frac{3}{8}x+5<26

3. 11<4-x

Answers

1. Plug -5 into the inequality.

-3(-5)+7 &>12\\15+7&>12

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

2. No negatives with the x- term, so we can solve this inequality like an equation.

& \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

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

6+5 < 26

The graph looks like:

3. In this inequality, we have a negative x- term. Therefore, we will need to flip the inequality.

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

Test a solution, x = -10: 11 < 4-(-10) \checkmark

11 < 14

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

Vocabulary

Inequality
A mathematical statement that relates two expressions that are not equal.

Practice

Solve and graph each inequality.

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

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

Image Attributions

Reviews

Email Verified
Well done! You've successfully verified the email address .
OK
Please wait...
Please wait...
ShareThis Copy and Paste

Original text