Question 510·Hard·Algebra

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$2x + 3y < 9$

$x - y \ge 1$

Which of the following represents the set of all $y$ -coordinates of points $(x,y)$ that satisfy the system of inequalities above?

A system of inequalities gives conditions that must hold at the same time. When a question asks for every possible $y$ -coordinate, graph both conditions in Desmos to see their overlap. Then find its upper limit exactly: for each $y$ , use the smallest $x$ the system allows. Watch whether a strict inequality excludes the boundary.

Hints

Hint 1
The feasible region is the part of the graph shaded by both inequalities. Look toward its highest edge. Which boundaries meet there, and does the strict inequality allow points on that edge?
Hint 2
The condition $x-y\ge1$ gives a lower bound on $x$ : for each $y$ , $x$ cannot be smaller than $y+1$ . Why does that smallest $x$ give the other inequality its best chance to hold?
Hint 3
After choosing the smallest $x$ , solve the resulting one-variable inequality for $y$ . Check that every $y$ below the bound works with that choice of $x$ , not only that values above it fail.

Step-by-step

Graph the overlap, then find its exact upper edge
Step 1
See where both conditions hold
Type $2x+3y<9$ and $x-y\ge1$ on separate Desmos lines. The overlap is shaded by both, and it narrows upward toward where their boundaries meet. The boundary for $<$ is dashed, so points on it do not count.
Step 2
Find the smallest allowed x
Add $y$ to both sides of $x-y\ge1$ : $x\ge y+1$ . So, for any fixed $y$ , $y+1$ is the smallest allowed $x$ . This is a lower bound: $x$ may be larger, but it cannot be smaller.
Step 3
Test whether that x can satisfy the other inequality
The term $2x$ grows when $x$ grows. So if the smallest allowed $x$ fails $2x+3y<9$ , every larger $x$ fails too. For each $y$ , test the smallest allowed $x$ . Substitute $x=y+1$ : $2(y+1)+3y<9$ . Distribute: $2y+2+3y<9$ . Combine like terms: $5y+2<9$ . Subtract $2$ : $5y<7$ . If this holds, $x=y+1$ satisfies both inequalities; if it fails, no $x$ can.
Step 4
State the possible y-coordinates
Divide by the positive $5$ , which keeps the inequality pointing the same way: $y<\frac{7}{5}$ . Every $y$ below this value works with $x=y+1$ . The upper edge does not count because $2x+3y<9$ is strict. So this describes all possible $y$ -coordinates. Choice D.

Question 510·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this