Question 102·Hard·Linear Inequalities in One or Two Variables
Which of the following represents the set of all -coordinates of points that satisfy the system of inequalities above?
For systems of linear inequalities where they ask for all possible x- or y-values, rewrite each inequality to bound the variable you are not asked about in terms of the variable you are asked about (here, solve for x in terms of y). For each fixed y, require that there is at least one value of the other variable that satisfies all inequalities—this becomes a single inequality in the desired variable. Solve that inequality carefully, paying attention to strict (<, >) versus non-strict (≤, ≥) signs, and then match the resulting range (such as y < k) to the answer choices.
Hints
Focus on y, not x
You are not being asked for all (x, y) pairs, only for all possible y-values. Think about, for each fixed y, whether there exists at least one x that can satisfy both inequalities at the same time.
Rewrite the inequalities in a helpful form
Try solving each inequality for x in terms of y so you can see, for a given y, what range of x-values is allowed by each inequality.
Combine the x-ranges
Once you have an upper and lower bound for x in terms of y, ask: what condition must those bounds satisfy so that there is some x between them? Turn that into an inequality in y and solve it.
Desmos Guide
Graph the boundary lines
In Desmos, type 2x + 3y = 9 and x - y = 1 as two separate equations. These will draw the boundary lines for the inequalities.
Find the intersection point of the lines
Click on the point where the two lines intersect; Desmos will display its coordinates. Note the y-coordinate of this point—that value represents the highest y-value at the edge of the overlapping region.
Check the shading directions for the inequalities
Now enter the actual inequalities: 2x + 3y < 9 and x - y >= 1. Desmos will shade the solution region. Confirm that the region lies below the first line (because of <) and on or above the second line (because of >=). The largest y-value that appears in this shaded region is just below the y-coordinate you found at the lines' intersection, so the correct answer choice is the one stating that y is less than that number.
Step-by-step Explanation
Express each inequality as a condition on x for a fixed y
We want all possible y-coordinates for points (x, y) that satisfy both inequalities.
Start by solving each inequality for x in terms of y:
From :
From :
So, for a given y, any valid x must satisfy both and .
Require that some x-values actually work for that y
For a y-value to be allowed, there must be at least one x that fits between the lower and upper bounds.
The lower bound on x is (from ).
The upper bound on x is (from ).
For there to be any x at all, the lower bound must be strictly less than the upper bound:
It must be strict because has to be the lower bound and the upper bound; if they were equal, there would be no number that is both and that same value.
Solve the inequality for y to find the allowed y-values
Now solve
First clear the fraction by multiplying both sides by 2:
Add to both sides:
Subtract 2 from both sides:
Now divide both sides by 5 (a positive number, so the inequality direction stays the same):
This means all possible y-coordinates of points satisfying the system are exactly the real numbers less than , which corresponds to choice D.