Systems of linear inequalities: shade both, read the overlap

An original SAT-style practice question on a system of linear inequalities, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with the substitute-and-solve method alongside.

Practice question

y < 3x − 2
y ≥ −2x + 6

In the xy-plane, the point (4, k) is a solution to the system of inequalities above. What is the greatest possible integer value of k?

⚡ Desmos makes this one hard to get wrong It shades each inequality and draws a strict boundary (< or >) as a dashed line, so you can see both the overlap and which edge is left out.

Linear inequalitiesSystemsMedium⏱ ~45 sec

Step by step

Type both inequalities as given

Desmos graphs inequalities directly. Type < as-is, and type >= or <= to get ≥ or ≤ (Desmos converts them as you type).

y<3x-2Blue shading, dashed boundary

y>=-2x+6Shows as y ≥ −2x + 6; red shading, solid boundary

x=4Vertical line for the point's x-value

What you see: two shaded half-planes. Where they overlap, the colours stack and look darker — that wedge, opening to the right, is the solution region. The vertical line x = 4 runs through it.

Find where x = 4 leaves the overlap

Follow the vertical line up through the darker region. It enters at the solid red line and leaves at the dashed blue line. Both crossings land on gridlines, so you can read them off; if you want to be sure, type the boundary values on new lines.

3(4)-2= 10, the top edge (dashed)

-2(4)+6= −2, the bottom edge (solid)

What you see: along x = 4, the overlap runs from y = −2 up to y = 10. The top edge is dashed, so y = 10 itself is not in the region.

Check the candidates as points

Type the two candidates as points. (4, 9) sits inside the darker region; (4, 10) sits right on the dashed line, which isn't part of the solution.

(4,9)Inside the overlap ✓

(4,10)On the dashed line ✗

Answer

At x = 4 the system becomes −2 ≤ k < 10, so the greatest integer is 9. Check (4, 9): 9 < 3(4) − 2 = 10 ✓ and 9 ≥ −2(4) + 6 = −2 ✓.

⚠ Common trap

Answering 10. That's where x = 4 meets the boundary y = 3x − 2, but the inequality is strict (<), so the boundary isn't included: 10 < 10 is false. Dashed line means the edge is out; solid line means it's in. Answering −2 is the other slip: that's the least possible value of k, not the greatest.

The algebra way, if you prefer

Substitute x = 4 into both inequalities: the first gives k < 3(4) − 2, so k < 10. The second gives k ≥ −2(4) + 6, so k ≥ −2. Together, −2 ≤ k < 10. The integers that work are −2 through 9, and the greatest is 9.

That's quick once you think to substitute. Graphing earns its keep on the versions where you have to pick which of several points is in the region, or the region's corner matters: solve the two boundary equations as a system (3x − 2 = −2x + 6 gives the corner at (1.6, 2.8)), or just tap where the lines meet on the graph.

More walkthroughs: systems of linear equations · solving for x by graphing · line of best fit · quadratic roots · vertex and maximum value · circle equations · exponential growth · sum of a quadratic's solutions · absolute value equations · Desmos cheat sheet · all walkthroughs

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