Question 18·Hard·Linear Inequalities in One or Two Variables
A tutoring service charges a one-time enrollment fee of dollars plus dollars for each hour of tutoring. This month, Alex wants the total amount he spends to be at least times and at most times the enrollment fee.
Let represent the number of tutoring hours Alex purchases this month.
Solving for , which inequality represents all possible values of that satisfy Alex’s goal?
For word problems that describe a total cost being between two multiples of a fee or base amount, first write an expression for the total cost, then place that expression in the middle of a compound inequality with the given lower and upper bounds. Solve the three-part inequality just like an equation: apply each operation (such as subtracting a term or dividing by a constant) to all three parts, and make sure the final answer has the requested variable isolated in the middle; be wary of choices that match only an intermediate step rather than the fully solved inequality.
Hints
Express the total cost in terms of f and h
Alex pays the enrollment fee once and then 18 dollars for each hour. How can you write his total monthly cost using f and h?
Turn the word description into an inequality
The phrase "at least 1.5 times and at most 2 times the enrollment fee" describes a range. What expression should go in the middle of that range: 1.5f ___ 2f?
Use a compound inequality and solve step-by-step
Once you have something like (total cost) , treat it like a three-part inequality: first subtract from all three parts, then divide all three parts by 18 so that h is by itself.
Check what the final inequality should look like
In the final answer, h should be alone in the middle of the inequality, with expressions involving f on both sides. Any option that still has 18h or f + 18h in the middle is not fully solved for h.
Desmos Guide
Set a value for the enrollment fee
In a Desmos expression line, assign a positive value to f, such as f = 60. Any positive number works; you just need a concrete value to visualize the relationships.
Graph the total cost as a function of hours
In a new line, type C(x) = f + 18x so that C(x) represents the total cost when x is the number of hours (x corresponds to h). This will graph a straight line increasing with x.
Graph the lower and upper spending limits
On two new lines, type y = 1.5f and y = 2f. These are horizontal lines showing 1.5 times and 2 times the enrollment fee. Look at where the line y = C(x) lies between these two horizontals and note the approximate x-values at the left and right intersection points.
Compare each answer choice to the graph
For each option, replace h with x and use your chosen f. For example, for choice A type 0.5f <= 18x <= f and see which x-values are shaded. The correct answer is the choice whose shaded x-interval exactly matches the range of x where the total cost line is between the two horizontal lines from the previous step.
Step-by-step Explanation
Write an expression for the total cost
Alex pays the one-time enrollment fee plus 18 dollars for each hour.
So his total cost this month, in dollars, is
Translate the word condition into a compound inequality
Alex wants the total he spends to be at least 1.5 times the enrollment fee and at most 2 times the enrollment fee.
That means the total cost must lie between and (including the endpoints):
This is the starting compound inequality before solving for .
Isolate the term that has h
To solve a three-part inequality, do the same operation to all three parts.
First, subtract from every part to move the enrollment fee to the left side:
Simplify each part:
Now the inequality is closer to being solved, but is still multiplied by 18.
Solve for h and match the answer choice
To get alone in the middle, divide every part of the inequality by 18:
This simplifies to
Since , we have
So the final inequality is
which matches answer choice B.