Inequalities and shaded overlap

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Difficulty: Beginner
Estimated time: 35 minutes
Domains: Algebra
Techniques: InequalitiesRegions

What you’ll learn

Enter linear inequalities directly in Desmos.
Interpret a shaded half-plane as a solution set.
Distinguish solid included boundaries from dotted excluded boundaries.
Test ordered pairs against one inequality.
Find the overlap that satisfies every inequality in a system.
Translate simple contextual limits into inequalities.
Choose substitution when checking one point by hand is faster.

Prerequisites

Why this matters on the SAT

Find the point that satisfies every inequality

SAT inequality questions often give several conditions and ask which ordered pair makes all of them true. Enter the inequalities directly in Desmos. Each one shades its solution region, and the system’s solutions are the points where every shading overlaps.

This extends the shared-solution idea from systems at intersections. Equations share points on their graphs. Inequalities share an entire region.

SAT example

Which ordered pair $(x,y)$ satisfies the system of inequalities below?

\begin{aligned} y&>2x-3\\ x+y&\le6 \end{aligned}

A
$(0,-3)$
B
$(2,2)$
C
$(3,3)$
D
$(4,-1)$

Fast Desmos solution

Enter both inequalities exactly as written, then plot the four answer-choice points. Only $(2,2)$ lies in both shaded regions.

The points $(0,-3)$ and $(3,3)$ lie on the dotted boundary of $y>2x-3$ , so they are excluded. The answer is B.

Desmos makes the full solution region visible at once. You still need to check boundary inclusion, coordinate order, and any restrictions from the situation.

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A system is satisfied only in the region where every inequality is shaded.

When should you graph inequalities in Desmos?

Desmos is strongest when several two-variable conditions must be true at the same time and the graph makes their shared region visible.

Begin with Desmos when…

the problem contains inequalities in both $x$ and $y$ ;
two or more inequalities must be satisfied at the same time;
several ordered-pair choices must be tested against multiple or awkward inequalities;
the boundary forms or coefficients make mental graphing slow; or
a context describes several limits on two quantities.

Use substitution when…

the question gives one simple inequality and one point to check. For $(2,1)$ and $3x+y\le8$ , one calculation gives $3(2)+1=7\le8$ .

Look for wording such as “Which ordered pair satisfies the inequality?”, “Which point is a solution to the system?”, “Which combination could meet all the requirements?”, or “Which point lies in the solution region?”

Check your understanding:

Which is a stronger reason to begin with Desmos: checking whether $(2,1)$ satisfies $3x+y\le8$ , or finding which point satisfies three two-variable inequalities? Explain the difference.

1. Read the shading and the boundary

An inequality divides the coordinate plane into:

a boundary line, where the two sides are equal; and
a shaded half-plane, containing the points that make the inequality true.

For

y\le-\frac{1}{2}x+4,

Desmos shades below the line. The boundary is solid because equality is allowed.

$(2,3)$ lies on the boundary and is included;
$(2,1)$ lies in the shaded region; and
$(2,5)$ lies outside the shaded region.

Inequality	Boundary	Included?
$<$ or $>$	Dotted	No
$\le$ or $\ge$	Solid	Yes

Type <= or >= in Desmos to enter $\le$ or $\ge$ .

Try it yourself:

Change <= to <. Watch the boundary change, then decide what happens to the point $(2,3)$ .

Check your understanding:

For $y<2x+1$ , does the point $(1,3)$ satisfy the inequality?

Common mistake:

Assuming a point on the boundary always works. It works only when the inequality includes equality.

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The solid line is included; points below it are also solutions.

2. Plot the choices, then verify

Desmos can graph an inequality in its original form. You do not need to solve it for $y$ first.

Suppose the question asks which point satisfies

4x+3y\le18.

Use this workflow:

Enter 4x+3y<=18.
Plot each answer-choice point on its own line.
Keep a point only if it lies in the shaded region or on an included boundary.
Substitute the final candidate into the inequality to verify it exactly.

Check your understanding:

Which plotted point satisfies $4x+3y\le18$ : $(0,6)$ , $(2,4)$ , $(3,3)$ , or $(4,1)$ ?

Common mistake:

Swapping the coordinates. In $(2,4)$ , substitute $x=2$ and $y=4$ , in that order.

If a point looks extremely close to a boundary, do not trust the picture alone. Substitute its coordinates to confirm whether the inequality is true.

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Plot candidate points without rearranging the inequality.

3. Example: Find the overlapping region

A point satisfies a system only if it belongs to every shaded region at the same time. A point in just one shading is not a solution to the system.

Worked example

Which ordered pair $(x,y)$ satisfies the system of inequalities below?

x>0

y\ge x+1

y<-2x+7

A
$(1,1)$
B
$(2,3)$
C
$(1,3)$
D
$(3,0)$

Step 1

Keep every condition visible

The point must be to the right of $x=0$ , on or above $y=x+1$ , and below $y=-2x+7$ .

Step 2

Graph the full system

Enter all three inequalities on separate lines. Their shared solution is the region where all three shadings overlap.

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The solution must lie in all three shaded regions.

Step 3

Test the candidate points

Choice C, $(1,3)$ , lies inside the complete overlap. Choice A is below the solid boundary, choice B lies on the dotted boundary, and choice D is below the required region.

Step 4

Verify the answer

For $(1,3)$ ,

1>0,\qquad 3\ge1+1,\qquad 3<-2(1)+7.

All three statements are true, so the answer is C.

Common mistake:

Choosing a point that lies in two shaded regions but not the third. Every inequality must be true.

This lesson uses overlap to test points. Questions that ask which parameter creates no solutions or at least one solution require the later slider and solution-count workflows.

4. Translate limits into a feasible region

In a context, each inequality represents one limit. The overlapping region contains every combination that obeys all the limits.

Suppose a school buys $x$ notebooks for $1 each and $y$ folders for $1 each. It needs at least $24$ items and can spend at most $1.

Check your understanding:

Before looking below, write the quantity inequality and the budget inequality.

The full system is

\begin{aligned} x+y&\ge24\\ 2x+3y&\le60\\ x&\ge0\\ y&\ge0. \end{aligned}

The point $(12,12)$ lies in the overlap. It represents $24$ items costing exactly $1.

Check your understanding:

Why does $(12,12)$ satisfy the school’s constraints while $(6,18)$ does not?

Common mistake:

Treating every visible point in the overlap as realistic. Counts must also be nonnegative whole numbers, even though the graph shades fractional points.

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The overlap shows combinations that meet the quantity and budget limits.

Finish the solution

The inequalities are already graphed. Plot the answer choices, find the point in the overlap, and submit its letter.

Find the point in the overlap

Finish the solution

Which ordered pair $(x,y)$ satisfies the system of inequalities below?

\begin{aligned} y&>-x+2\\ y&\le2x+5 \end{aligned}

First steps

Both inequalities are entered and their shaded regions are visible.
Plot each answer-choice point on its own expression line.
Keep only a point that lies in both shadings and respects the boundary style.

Finish it

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The inequalities are ready. Plot each choice and inspect the overlap.

Practice problems

SAT practice problems

For stated inequalities, choose graphing when several conditions make the comparison repetitive and substitution when one short check is faster. For contexts, translate every condition first and map the quantities to $x$ and $y$ when Desmos needs graphing variables. Verify the final point by substitution.

Test one inequality

Practice problem

Which ordered pair $(x,y)$ satisfies $2x+5y\le16$ ?

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Find the shared region

Practice problem

Which ordered pair $(x,y)$ satisfies the system of inequalities below?

\begin{aligned} y&>x-2\\ 2x+y&\le8 \end{aligned}

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Test contextual constraints

Practice problem

A theater sells adult tickets for $1 each and student tickets for $1 each. For one show:

at most $24$ tickets can be sold;
ticket revenue must be at least $1; and
at least $8$ student tickets must be sold.

Let $a$ be the number of adult tickets sold and $s$ be the number of student tickets sold. Which ordered pair $(a,s)$ could represent the ticket sales?

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Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Enter inequalities directly; do not rearrange them unless that is clearly faster.
The shaded region contains the points that make an inequality true.
Dotted boundaries are excluded; solid boundaries are included.
A system’s solutions lie in every shaded region at the same time.
Plot ordered-pair choices, then verify the final candidate by substitution.
In context, translate every limit and keep count restrictions such as nonnegative whole numbers.
Use hand substitution when checking one point takes only one short calculation.

Next lesson

Sliders for unknown constants

Vary an unknown constant and observe how it changes an equation, function, or graph.

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Free Practice

Free question bank practice problems

Linear inequalities in one or two variables