1.9: Checking Solutions to Inequalities
What if you were given an inequality like \begin{align*}3x^3 < 81\end{align*}
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CK12 Foundation: 0109S Check Solutions to Inequalities
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For more practice solving inequalities, check out http://www.aaastudy.com/equ725x7.htm.
Guidance
To check the solution to an inequality, we replace the variable in the inequality with the value of the solution. A solution to an inequality produces a true statement when substituted into the inequality.
Example A
Check that the given number is a solution to the inequality: \begin{align*}a = 10; \ 20a \le 250\end{align*}
Solution
Replace the variable in the inequality with the given value.
\begin{align*}20(10) & \le 250\\ 200 & \le 250\end{align*}
This statement is true. This means that \begin{align*}a = 10\end{align*}
Note that \begin{align*}a = 10\end{align*}
Example B
Check that the given number is a solution to the inequality: \begin{align*}b = 0.5; \ \frac{3  b}{b} > 4\end{align*}
Solution:
\begin{align*} \frac{3  (0.5)}{(0.5)} & > 4\\ \frac{3 + 0.5}{0.5} & > 4\\  \frac{3.5}{0.5} & > 4\\ 7 & > 4\end{align*}
This statement is false. This means that \begin{align*}b =  0.5\end{align*}
Example C
To organize a picnic Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs. What is the possible number of hamburgers Peter has?
Solution:
Define
Let \begin{align*}x =\end{align*}
Translate
Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs.
This means that twice the number of hot dogs is less than or equal to the number of hamburgers.
\begin{align*}2 \times 24 \le x, \ \text{or} \ 48 \le x\end{align*}
Answer
Peter needs at least 48 hamburgers.
Check
48 hamburgers is twice the number of hot dogs. So more than 48 hamburgers is more than twice the number of hot dogs. The answer checks out.
Watch this video for help with the Examples above.
CK12 Foundation: Check Solutions to Inequalities
Vocabulary
 A solution to an inequality produces a true statement when substituted into the inequality.
Guided Practice
Check that the given number is a solution to the inequality: \begin{align*}x = \frac{3}{4}; \ 4x + 5 \le 8\end{align*}
Solution:
\begin{align*}4 \left ( \frac{3}{4} \right ) + 5 & \ge 8\\ 3 + 5 & \ge 8\\ 8 & \ge 8\end{align*}
This statement is true. It is true because this inequality includes an equals sign; since 8 is equal to itself, it is also “greater than or equal to” itself. This means that \begin{align*}x = \frac{3}{4}\end{align*}
Practice
For 14, check whether the given number is a solution to the corresponding inequality.

\begin{align*}x = 12; \ 2(x + 6) \le 8x\end{align*}
x=12; 2(x+6)≤8x 
\begin{align*}z = 9; \ 1.4z + 5.2 > 0.4z\end{align*}
z=−9; 1.4z+5.2>0.4z 
\begin{align*}y = 40; \  \frac{5}{2}y + \frac{1}{2} < 18\end{align*}
y=40; −52y+12<−18 
\begin{align*}t = 0.4; \ 80 \ge 10(3t + 2)\end{align*}
t=0.4; 80≥10(3t+2)  On your new job you can be paid in one of two ways. You can either be paid $1000 per month plus 6% commission of total sales or be paid $1200 per month plus 5% commission on sales over $2000. For what amount of sales is the first option better than the second option? Assume there are always sales over $2000.
For 614, suppose a phone company offers a choice of three textmessaging plans. Plan A gives you unlimited text messages for $10 a month; Plan B gives you 60 text messages for $5 a month and then charges you $0.05 for each additional message; and Plan C has no monthly fee but charges you $0.10 per message.
 If \begin{align*}m\end{align*}
m is the number of messages you send per month, write an expression for the monthly cost of each of the three plans.  For what values of \begin{align*}m\end{align*}
m is Plan A cheaper than Plan B?  For what values of \begin{align*}m\end{align*} is Plan A cheaper than Plan C?
 For what values of \begin{align*}m\end{align*} is Plan B cheaper than Plan C?
 For what values of \begin{align*}m\end{align*} is Plan A the cheapest of all? (Hint: for what values is A both cheaper than B and cheaper than C?)
 For what values of \begin{align*}m\end{align*} is Plan B the cheapest of all? (Careful—for what values is B cheaper than A?)
 For what values of \begin{align*}m\end{align*} is Plan C the cheapest of all?
 If you send 30 messages per month, which plan is cheapest?
 What is the cost of each of the three plans if you send 30 messages per month?
inequality
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .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.Image Attributions
Description
Learning Objectives
Here you'll learn to check that a given number is a solution to an inequality.