Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Inequalities from Word Problems on Desmos (Digital SAT)

Desmos shades the allowed values, which confirms the direction once the expression is built. Turning the words into a rate, a start, and the right symbol is the reading you do first.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for inequalities from word problems

The Desmos steps

  1. 1

    Graph the inequality

    Type your inequality into Desmos to see the shaded region of allowed values.

  2. 2

    Check a known case

    Confirm a value that should work lands in the shaded region, and one that should not lands outside.

  3. 3

    Compare choices

    Graph each and keep the one whose region matches the situation.

Worked in Desmos

Easy example
A saver starts with $200 and adds $25 each week. To have at least $500 after ww weeks, which inequality represents this situation?
200+25w≥500200 + 25w \geq 500
B
200+25w≤500200 + 25w \leq 500
C
25w≥50025w \geq 500
D
200+25w>500200 + 25w > 500

A: Correct. The balance 200+25w200 + 25w must be at least 500: 200+25w≥500200 + 25w \geq 500.

B: Incorrect. "At least" limits the balance below, so use ≥\geq, not ≤\leq.

C: Incorrect. This drops the initial $200.

D: Incorrect. "At least" includes the boundary, so use ≥\geq, not strict >>.

Explanation

Balance: start plus weekly =200+25w= 200 + 25w. "At least $500" means 200+25w≥500200 + 25w \geq 500.

Medium example
A phone plan charges a $30 monthly fee plus $0.10 per text message. A customer wants the monthly bill to stay at most $50. Which inequality represents the possible numbers of text messages tt?
A
30+0.10t≥5030+0.10t\geq50
30+0.10t≤5030+0.10t\leq50
C
0.10t≤500.10t\leq50
D
30t+0.10≤5030t+0.10\leq50

A: Incorrect. This reverses the inequality direction; "at most" requires ≤\leq, not ≥\geq.

B: Correct. The bill is the fee plus the per-text cost, 30+0.10t30+0.10t, and this must stay at most 50: 30+0.10t≤5030+0.10t\leq50.

C: Incorrect. This omits the $30 monthly fee.

D: Incorrect. This swaps which quantity is multiplied by tt, attaching the fee to tt instead of the per-text rate.

Explanation

The bill is 30 + 0.10t, and this must stay at most 50: 30 + 0.10t <= 50.

Hard example

A manufacturer packages items in boxes that hold 12 items each. A company needs to package between 500 and 600 items, inclusive, using only these boxes.

Which inequality represents the possible number of 12-item boxes nn needed to hold a total item count within this range?
500≤12n≤600500\leq12n\leq600
B
500≤n≤600500\leq n\leq600
C
12(500)≤n≤12(600)12(500)\leq n\leq12(600)
D
500<12n<600500<12n<600

A: Correct. Each box holds 12 items, so the total items for nn boxes is 12n12n. Setting up the inclusive range: 500≤12n≤600500\leq12n\leq600.

B: Incorrect. This forgets that nn represents boxes, not items, omitting the multiplier of 12: 500≤n≤600500\leq n\leq600.

C: Incorrect. This inverts the relationship, multiplying the bounds by 12 instead of setting up the total correctly: 12(500)≤n≤12(600)12(500)\leq n\leq12(600).

D: Incorrect. This uses strict inequalities instead of inclusive ones -- "between...inclusive" means the endpoints count: 500<12n<600500<12n<600.

Explanation

Each box holds 12 items, so the total for n boxes is 12n. The inclusive range is 500 <= 12n <= 600.

Try it with the calculator

Question 1easy
Shaded region in the xy-plane
-6-4-2246-6-4-2246O

The shaded region shown in the xy-plane represents the solutions to which of the following inequalities?

Question 2easy
Shaded region in the xy-plane
-6-4-2246-2246810O

The shaded region shown in the xy-plane represents the solutions to which of the following inequalities?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Desmos questions

Can Desmos solve inequalities from word problems questions on the SAT?

Desmos shades the allowed values, which confirms the direction once the expression is built. Turning the words into a rate, a start, and the right symbol is the reading you do first.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master inequalities from word problems, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full methodTake the free diagnostic