Inequalities That Shade Themselves: Regions on Desmos

Test the point, flip the sign, shade above or below — or type the inequality and let the calculator shade the truth for you.

The move: Type the inequalities, read the overlap.
Why it works: Desmos shades every point satisfying the inequality, so “which point satisfies both?” becomes “which choice sits in the double-shaded overlap?”

The pattern

1. Type the first inequality
2. Type the second — the overlap shades
3. Find the point inside the shaded overlap

Worked example: Which point satisfies BOTH y ≤ 2x + 1 and y > −x + 4?

1. Type the first inequality — Desmos shades everything that works. y ≤ 2x+1
2. Add the second. The overlap of the two shadings is where BOTH are true. y>-x+4
3. Now plot the answer choices as points and see which lands in the double-shaded zone: (3, 2). (0,0), (3,2), (0,5), (-2,1)

Worked example 2: How many INTEGER values of x satisfy |x| < 3.5?

1. Type the inequality in one variable — Desmos shades a vertical band. |x|<3.5
2. The band runs from −3.5 to 3.5 (dashed edges: endpoints excluded).
3. Count integers inside: −3, −2, −1, 0, 1, 2, 3 → seven.

Remember it on test day

Shade, then overlap. That's the whole cue — say it when you see this question type and the steps follow.

When this isn't the move

Strict vs inclusive boundaries (dashed vs solid) matter when a choice sits exactly on a line — check the symbol before trusting the picture.

Drill it until it's reflex

This move is one of Desvantage's 18 training modules: a plain-English lesson, a type-it-yourself walkthrough inside a live Desmos calculator, and a 100-question drill bank that deals ten fresh questions per round. The first two modules — plus the 12-question diagnostic — are free.