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Checking Solutions to Inequalities

Substitute values for variables to check inequality solutions

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Checking Solutions to Inequalities

Checking Solutions: Inequalities 

To check the solution to an inequality, replace the variable in the inequality with the value of the solution. If the solution is correct, the simplified inequality will produce a true statement.

Checking Solutions to Inequalities

1. Check that the given number is a solution to the inequality: \begin{align*}a = 10; \ 20a \le 250\end{align*}a=10; 20a250

Replace the variable in the inequality with the given value.

\begin{align} 20(a) & \le 250\\ 20(10) & \le 250\\ 200 & \le 250\\ \end{align}20(a)20(10)200250250250

\begin{align*}200 \le 250\end{align*}200250 is a true statement. This means that \begin{align*}a = 10\end{align*}a=10 is a solution to the inequality \begin{align*}20a \le 250\end{align*}20a250.

Note that \begin{align*}a = 10\end{align*}a=10 is not the only solution to this inequality. Recall that inequalities may be simplified using essentially the same processes as equations. If we divide both sides of the inequality by 20, we can write it as \begin{align*}a \le 12.5\end{align*}a12.5. This means that actually any number less than or equal to 12.5 is a solution to the inequality.

2. Check that the given number is a solution to the inequality: \begin{align*}b = -0.5; \ \frac{3 - b}{b} > -4\end{align*}b=0.5; 3bb>4

\begin{align} \frac{3 - (b)}{(b)} & > -4\\ \frac{3 - (-0.5)}{(-0.5)} &> -4\\ \frac{3 + 0.5}{-0.5} &> -4\\ - \frac{3.5}{0.5} & > -4\\ -7 &> -4\\ \end{align}3(b)(b)3(0.5)(0.5)3+0.50.53.50.57>4>4>4>4>4

This statement is false. This means that \begin{align*}b = - 0.5\end{align*}b=0.5 is not a solution to the inequality \begin{align*}\frac{3 - b}{b} > -4\end{align*}3bb>4.

Real-World Application 

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?

Define

Let \begin{align*}x\end{align*}x represent the number of hamburgers

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*}2×24x, or 48x

Answer

Peter needs at least 48 hamburgers.

Check

48 hamburgers is twice the number of hot dogs. More than 48 hamburgers is more than twice the number of hot dogs. The answer checks out.

Example

Example 1

Check that the given number is a solution to the inequality: \begin{align*}x = \frac{3}{4}; \ 4x + 5 \le 8\end{align*}x=34; 4x+58

\begin{align} 4(x)+5 &\le 8\\ 4 \left ( \frac{3}{4} \right ) + 5 &\ge 8\\ 3 + 5 &\ge 8\\ 8 &\ge 8\\ \end{align}

The statement \begin{align*}8 \ge 8\end{align*} is true because this inequality includes an equals sign, and 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*} is a solution to the inequality \begin{align*}4x + 5 \le 8\end{align*}.

Review

For 1-4, check whether the given number is a solution to the corresponding inequality.

  1. \begin{align*}x = 12; \ 2(x + 6) \le 8x\end{align*}
  2. \begin{align*}z = -9; \ 1.4z + 5.2 > 0.4z\end{align*}
  3. \begin{align*}y = 40; \ - \frac{5}{2}y + \frac{1}{2} < -18\end{align*}
  4. \begin{align*}t = 0.4; \ 80 \ge 10(3t + 2)\end{align*}
  5. 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 6-14, suppose a phone company offers a choice of three text-messaging 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.

  1. If \begin{align*}m\end{align*} is the number of messages you send per month, write an expression for the monthly cost of each of the three plans.
  2. For what values of \begin{align*}m\end{align*} is Plan A cheaper than Plan B?
  3. For what values of \begin{align*}m\end{align*} is Plan A cheaper than Plan C?
  4. For what values of \begin{align*}m\end{align*} is Plan B cheaper than Plan C?
  5. 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?)
  6. 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?)
  7. For what values of \begin{align*}m\end{align*} is Plan C the cheapest of all?
  8. If you send 30 messages per month, which plan is cheapest?
  9. What is the cost of each of the three plans if you send 30 messages per month?

Review (Answers)

To view the Review answers, open this PDF file and look for section 1.9. 

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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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