Desmos for the SAT · Technique 17 of 23

Solve an Inequality by Graphing

Type it as printed. Desmos shades every value that works, and the edge tells you the rest.

SolvesSolve an inequality for xWhich values of x satisfy this?The solution set of an inequalityThe least or greatest value that worksHow many integers satisfy it

Watch the lesson

The whole move worked on one real SAT question, narrated.

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One inequality typed as printed, the shaded side read off, and the boundary dot pinned.

Solving an inequality by hand runs exactly like solving an equation, with one extra rule: multiply or divide both sides by a negative number and the symbol has to turn around. When that turn goes missing, every line after it is right about the number and wrong about which side of it you want, which is why the SAT prints the turned-around version as one of the four choices.

Graphing removes the rule. Type the inequality into Desmos exactly as it is printed, and it shades every value of x that makes the statement true. Nothing is divided, so there is no rule to apply: you are looking at which side of a vertical boundary the shading sits on.

Two things to read off the picture. Which side is shaded tells you greater than or less than. Whether that edge, the boundary, is dashed or solid tells you whether the boundary value itself counts, and that is the whole difference between "greater than" and "greater than or equal to" on the answer sheet.

This is the inequality version of solving an equation by graphing. There, two graphs cross at a point and you read the x. Here, one line shades a band and you read its edge. If a question gives you two inequalities and four points instead, that is graphing a system of inequalities. If the four choices are inequalities describing a word problem, and they disagree about more than the symbol, that is which inequality represents this situation.

Nova

The by-hand route has one step where a sign decides everything, and the page gives you no way to check it. Typing the inequality skips that step. Desmos shades the values that work, you read the side and the edge, and the boundary number comes off a dot you click. This lesson is about reading that picture fast.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps clearing the fractions and then dividing by a negative.

The question

$\frac{5x-3}{4}\ge\frac{3x+1}{2}-2$

How many positive integers satisfy it?

The algebra way

Clear the denominators by multiplying every term by 4: 5x - 3 >= 2(3x + 1) - 8.
Expand and tidy up: 5x - 3 >= 6x - 6.
Collect the x terms on one side: -x >= -3.
Divide by -1, which reverses the symbol, giving x <= 3. Leave it pointing the way it was and the answer comes out as every integer from 3 upward.

The Desmos way

Type the inequality on one line, with x in place of n, exactly as printed.
Read the shading: a band filling everything left of a solid edge.
Click inside the box, click the dot on the edge, and read $(3,0)$ .

The payoff: The algebra ends on the one step where the symbol changes meaning, with no way to check that you made the change. The graph never gets there: the shaded side was decided the moment you finished typing, and it is the same picture whether the coefficient was positive or negative.

Quick reference

What the shading is telling you

What you see	What it means	How the answer reads
Shading to the right of the edge	Every value above the boundary works	$x>k$ or $x\ge k$
Shading to the left of the edge	Every value below the boundary works	$x<k$ or $x\le k$
A dashed edge	Strict inequality: the boundary value is not a solution	The symbol has no line under it
A solid edge	The boundary value is included	The symbol has a line under it
The whole plane shaded, no edge	Every real number satisfies it	"All real numbers", which is a real answer choice
Nothing shaded at all	No value of x satisfies it	"No solution"

Step by step

The whole move, all 6 steps

Type the inequality on one line, exactly as printed, using x as the variable. Swap the question's letter for x first, because Desmos only shades inequalities written in x and y.
Look at the shading rather than hunting for a curve. Desmos fills in every x that satisfies the inequality, so the solution set is a band of the plane with a vertical edge.
Read which side of that edge is shaded. Shading to the right means the answer is greater than the boundary value, shading to the left means less than.
Check whether the edge is dashed or solid. Dashed means the boundary value itself is out, and solid means it is in.
Click inside the expression box to reveal the gray dot where the boundary meets the x-axis, then click that dot to pin its coordinates. The x-coordinate is the number the solution set stops at.
When the answer choices are themselves inequalities, type one on the next line. The choice that shades exactly the same band is the answer, and a choice with the opposite symbol shades the opposite side instead.

The question

The inequality $\frac{5n-3}{4}\ge\frac{3n+1}{2}-2$ is given.

How many positive integers n satisfy the inequality?

A0
B1
C3
DInfinitely many

Step 1 of 4

The question is written in n, and Desmos shades in x, so swap the letter and type the whole thing on one line: (5x-3)/4 >= (3x+1)/2 - 2. Press the right arrow after each denominator to come back out of the fraction, and type > then = for the $\ge$ symbol.

In the calculator: (5x-3)/4→>=(3x+1)/2→-2

Nothing is rearranged and no denominators are cleared. Both sides go in the way the page prints them.

Now try it on a real SAT question

Head or scratch paper. Thirty miles at five miles an hour is one division, and you can say the answer out loud before the calculator has drawn an axis. Spotting the ones to read instead of type is worth as many points as the typing.

Maria set a goal to run at least 30 miles every week to train for a marathon. On a certain week, Maria plans to run at an average speed of 5 miles per hour.

What is the minimum number of hours Maria must run during that week to fulfill the weekly goal?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

Straight from Desmos

Desmos Help Center: InequalitiesHow Desmos shades inequalities, and what the dashed and solid boundaries mean.

The Desmos trap: when to close the calculator

Write it in x. Desmos only shades inequalities in x and y, so a question set in n, h or t has to have its letter swapped first. Typed as 5n >= 30 it refuses to plot and says so in the expression list.
Nothing is marked until the expression is selected. Click inside the expression box first, and the gray dot appears where the edge meets the x-axis. Hovering it shows the coordinates; it takes a click to pin them.
If the whole plane goes shaded, every real number satisfies the inequality, and if nothing shades at all, no value does. Both are answer choices on some questions, so read the blank screen as a result rather than as a typo.
When the question can be settled with one division, divide. Thirty miles at five miles an hour is faster to say than to type.
Two inequalities and four ordered pairs is a different question. That one is about the overlap of two shaded regions in the xy-plane, and it lives in graphing a system of inequalities.
Dashed or solid is part of the answer, not decoration. A strict symbol draws a dashed edge and the boundary value is out; a symbol with a line under it draws a solid edge and the boundary value is in. On a question whose four choices are the same number with four different symbols, that edge is the only thing separating them.

Nova

Check the sentence before you type, though. If it comes down to one division, like thirty miles at five miles an hour, do the division. The calculator is for the ones with fractions on both sides, or a negative sitting in front of the x, where the symbol could end up pointing either way.

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Put this move into real practice

"Solve an Inequality by Graphing" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

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