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(Videos) Solving Word Problems Involving Inequalities

by PatrickJMT

This video by Patrick JMT works through an inequality word problem. There is another video with a second example later in the explanation.

$\large{\text{Summary}}$

The first example is:

A widget factory has a fixed operating cost of $\$3{,}600$ per day plus costs $\$1.40$ per widget produced. If a widget sells for $\$4.20$, what is the least number of widgets that must be sold per day to make a profit?

To make a profit, the revenue received needs to be greater than the cost needed to make it. $$\text{Revenue}>\text{Cost}$$

Now, we want to create an expression for each of these. Let's let $x$ be the number of widgets produced.

For cost, we know there's a fixed cost of $\$3600$ and then $\$1.40$ for each widget. $$\text{Revenue}>3600+1.4x$$

For revenue, we receive $\$4.20$ for each widget. $$4.2x>3600+1.4x$$

Finally, we solve by getting $x$ to one side of the inequality, similar to how you'd do for an equation. $$\begin{align} 4.2x&>3600+1.4x\\ 4.2\color{#8925AE}{-1.4x}&>3600+1.4x\color{#8925AE}{-1.4x}\\ 2.8x&>3600\\ x&>1285.71 \end{align}$$

So, in order to make a profit, you'd need to sell over $1285$ widgets.

(Videos) Solving Word Problems Involving Inequalities

by PatrickJMT

Here's another example.

Jason makes and sells fishing poles. If he has costs totaling $\$12{,}000$ per year plus a cost of $\$4$ per pole, how many fishing poles must he produce to make a profit of $\$48{,}000$ in one year, if he sells the fishing poles for $\$28$ each?

We want the profit to be greater than or equal to $\$48{,}000$. $$\text{Profit}\ge 48000$$

We can also rewrite profit as the amount you make minus the amount you spend. $$\text{amount made}-\text{amount spent}\ge 48000$$

If he sells the poles for $\$28$ each, then that is the amount he makes. We can let $x$ be the number of poles. $$28x-\text{amount spent}\ge48000$$

We also know there's a fixed cost to make the poles for $\$12000$. Also, each pole costs $\$4$ to make. $$28x-(12000+4x)\ge48000$$

Finally, we can solve the inequality by getting $x$ alone on one side. $$\begin{align} 28x-(12000+4x)&\ge48000\\ 28x-12000-4x&\ge48000\\ 24x-12000&\ge48000\\ 24x-12000\color{#8925AE}{+12000}&\ge48000\color{#8925AE}{+12000}\\ 24x&\ge60000\\ x&\ge2500 \end{align}$$

To get here, you may want to review the distributive properties with inequalities and solving inequalities via grouping.

Jason will need to sell at least $2{,}500$ poles in order to make a profit of $\$48{,}000$.

Inequality Word Problems

The key to word problems is translating the given information into math. In this case, we need to translate word problems into statements of inequality.

Inequalities are denoted by words like "greater than" and "less than". In general, a comparison is made in problems with inequalities.

Let's look at two examples:

Example 1:

We need to convert this information into an inequality. We are trying to figure out how many ducks Chris can by while staying at or under budget. In other words, we want to keep the cost less than or equal to his budget. Sounds like a good place to use a $≤$ sign, doesn't it?

Each duck is $\$6.95$, so we'll be multiplying $6.95$ times the number of ducks (let's call the number of ducks $x$). Chris has to pay $\$9.99$ on top of what the ducks cost no matter what, so that doesn't get multiplied. With that in mind, we can write the equation:

$$6.95x + 9.99 \le 80$$

This inequality will allow us to find the value of $x$ (the number of rubber ducks) that allows Chris to stay at or below his budget of $80$ dollars.

All we need to do now is solve for $x$.

$$\begin{align} 6.95x + 9.99 &\le 80\\ 6.95x &\le 70.01\\ x &\le 10.07338...\\\ x &\le 10.1, \text{ after rounding} \end{align}$$

So, our solution tells us that Chris can buy up to (and including) $10.1$ rubber ducks. Because only whole rubber ducks are sold, we need to round our answer down to $10$. (We can't round up to $11$, because then $x$ (the number of ducks) wouldn't be less than or equal to $10.1$.)

So, Chris can buy $10$ ducks!

Example 2:

$$18 + 1.25x < 3.5x$$

This inequality shows us Offer A on the left and Offer B on the right. We want to know the number of miles ($x$) that allows Offer A to cost less than Offer B. That means we want $18 + 1.25x,$ the cost of Offer A, to be less than $3.5x,$ the cost of Offer B.

$$\begin{align} 18 + 1.25x &< 3.5x \\ 18 &< 2.25x \\ 8 &< x \\\ x &> 8 \end{align}$$

So, we can say that Offer A is a better deal for Miguel if he rides more than $8$ miles.

Solving Word Problems Involving Inequalities

Inequality word problems will always involve comparisons. In real life, you might use inequalities to decide which of two options is better. Look for comparison words to figure out which kind of symbol you should use. Look for rates to write out an equation.

This is a tricky one! To start off with, the comparison words are "no less than". This is because these words make us compare Billie's annual income with a goal ($\$20{,}000$).

Now, onto the symbol. "No less than" might make you think that you should use a less than symbol ($<$), but that "no" at the beginning means the symbol should be greater ($>$ or $\ge$). Since $\$20{,}000$ is not less than itself, we need to include it in our possible solutions. The "greater than or equal to" ($\ge$) symbol is the way to go.

The correct inequality is $18,000 +5\% \cdot{x}\ge{20,000}$ with $x$ as number of sales. Now, we can solve this. We see, $$\begin{align} 18,000+0.05x&\ge{20,000}\\ 18,000+0.05x-{\color{#8925AE}{18,000}}&\ge{20,000}-{\color{#8925AE}{18,000}}\\ 0.05x&\ge{2,000}\\ \frac{0.05x}{0.05}&\ge\frac{2,000}{0.05}\\ x&\ge{40,000} \end{align}$$

Here is what a graph of the inequality would look like.

So, Billie must make at least $40,000$ sales.