How To Graph Inequalities On Ti 84 Plus: Step-by-Step Guide

Ever tried to sketch a system of inequalities on a TI‑84 Plus and ended up with a squiggly mess that looks nothing like the textbook picture?
You’re not alone. Most students stare at the calculator’s screen, punch in a couple of symbols, and wonder why the “solution region” never shows up Small thing, real impact..

The short version is: the TI‑84 can plot inequalities, but you have to coax it a little. Below is the step‑by‑step guide that actually works, plus the pitfalls most guides skip Took long enough..

What Is Graphing Inequalities on the TI‑84 Plus

When we talk about graphing an inequality, we mean drawing the set of all (x, y) points that satisfy a relationship like y ≤ 2x + 3 or x² + y² > 4.
On a computer you’d just shade the region, but the TI‑84 is a graphing calculator, not a full‑blown CAS. It can plot the boundary curve—line, parabola, circle, whatever—then you tell it which side to shade.

In practice the calculator treats the inequality as a regular function, draws the curve, and then uses a “test point” to decide whether to fill the area above or below that curve. The trick is getting the right mode and the right test point.

The “Y=” Editor Is Your Playground

All the action lives in the Y= screen. But each line (Y1, Y2, …) can hold a function, a parametric equation, or a polar curve. For inequalities you’ll usually work in function mode (the default) The details matter here..

Why It Matters / Why People Care

Understanding how to graph inequalities on the TI‑84 isn’t just a homework hack Worth keeping that in mind..

• Visual learning – Seeing the shaded region helps you grasp solution sets, especially for systems where the overlap matters.
• Exam confidence – AP Calculus, SAT Math, and many college courses allow the TI‑84 on the test. If you can quickly shade a feasible region, you’ll earn partial credit even when you can’t solve algebraically.
• Time saver – Sketching by hand takes minutes; the calculator does it in seconds, freeing you for the next problem.

But when you skip the proper steps, you get a blank screen or a wrong region, and that’s where grades slip.

How It Works (or How to Do It)

Below is the full workflow, broken into bite‑size pieces. Grab your TI‑84, follow along, and you’ll be shading like a pro That's the part that actually makes a difference..

1. Set the Right Mode

1. Press [MODE].
2. Make sure Func is selected (not Param, Polar, or Seq).
3. Scroll down to “Plot1: On/Off” – leave it Off; we’ll use the graph itself, not individual plots.
4. Hit [2nd] → [QUIT] to exit.

2. Enter the Inequality as a Function

Suppose you want to graph y ≥ −½x + 4.

1. Press [Y=].
2. On Y1, type -0.5X+4.
3. Don’t type the “≥” sign—just the right‑hand side of the inequality.

If you have a non‑linear inequality, like x² + y² ≤ 9, you need to solve for y first:

Y1 = sqrt(9‑X²) (upper semicircle)
Y2 = -sqrt(9‑X²) (lower semicircle)

3. Choose the Correct “Shade” Setting

The TI‑84 doesn’t have a built‑in shading command, but you can simulate it with the “Test‑Point” feature Most people skip this — try not to..

1. Press [2nd] → [DRAW] (the [PRGM] key).
2. Select “Test‑Pt”.
3. The calculator will ask “Test Pt X=”. Choose a value that lies on the side you don’t want shaded. For our line y ≥ −½x + 4, pick a point below the line, say X=0.
4. It then asks “Test Pt Y=”. Give a y‑value that’s below the line at that x, e.g., Y=0.

Now the calculator knows “the region containing (0, 0) is outside the solution”.

4. Turn on the “Shade” Feature

1. Press [2nd] → [DRAW] again.
2. Choose “Shade”.
3. The screen will prompt “Shade? (Y/N)” – answer Y.
4. You’ll see a prompt “Which Y‑line?” – type the number of the function you entered (usually 1).

The TI‑84 will now fill the appropriate side of the curve. If the shading looks backwards, just repeat steps 3‑4 with a different test point on the opposite side.

5. Adjust Window Settings

A badly set window can make the shaded region disappear off‑screen Easy to understand, harder to ignore..

• Press [WINDOW].
• Set Xmin, Xmax, Ymin, Ymax to values that comfortably surround your curve.
• For a circle of radius 3, Xmin=-4, Xmax=4, Ymin=-4, Ymax=4 works well.

Hit [GRAPH] to see the final picture And it works..

6. Graphing a System of Inequalities

When you have more than one inequality, repeat steps 2‑4 for each function (Y1, Y2, Y3…). The calculator will shade each region separately, and the overlapping area will appear darker—exactly what you need for feasible‑region problems.

Common Mistakes / What Most People Get Wrong

• Putting the inequality sign in the Y= editor. The TI‑84 only accepts expressions, not relational operators.
• Using the wrong test point. If the test point lies on the boundary, the calculator can’t decide and may leave the region unshaded.
• Forgetting to solve for y. Trying to enter X^2+Y^2≤9 directly will throw a syntax error.
• Leaving Plot1 on. That draws points instead of the curve, obscuring the shading.
• Skipping window adjustments. A default window of -10 to 10 might cut off a parabola’s vertex, making the shaded region look incomplete.

Practical Tips / What Actually Works

• Pick simple test points. The origin (0, 0) works for most linear inequalities; for circles, use a point clearly inside or outside, like (0, 0) for a circle centered at the origin.
• Use the “Trace” button after shading to verify that a point you know is a solution is indeed in the darkened area.
• Store repeated functions. If you frequently graph the same inequality, press [2nd] → [STO>]; store the expression in a variable (e.g., A). Then recall with ALPHA A. Saves typing.
• Combine with the “Calc” menu to find intercepts. Press [2nd] → [TRACE], choose “Calc”, then “Zero” to locate where the boundary crosses the axes—handy for checking work.
• Reset shading if it looks wrong: go back to [2nd] → [DRAW], select “Shade”, answer N, then redo the whole test‑point routine.

FAQ

Q: Can the TI‑84 graph strict inequalities ( < or > ) or only ≤/≥?
A: Yes. The shading works the same; the only difference is that the boundary line itself is drawn solid for ≤/≥ and dashed for < / >. You control that by selecting the appropriate line style in the Y= screen Small thing, real impact..

Q: What if the inequality involves absolute values, like |x − 2| ≤ 3?
A: Rewrite it as two separate inequalities: -3 ≤ X‑2 ≤ 3. In the Y= editor, enter Y1 = X‑2+3 and Y2 = X‑2‑3. Then shade the region between the two lines using test points on each side.

Q: My calculator says “ERROR: INVALID INPUT” when I type sqrt(9‑X²).
A: The TI‑84 requires the radicand to be non‑negative for real‑valued functions. Make sure you’re in function mode and that you’ve entered the expression exactly as sqrt(9‑X^2). Also check that the window includes the domain where the expression is defined But it adds up..

Q: Is there a way to save a shaded graph for later use?
A: Yes. After you’ve set everything up, press [2nd] → [MEM], choose “Save…”, then pick a slot in “Y‑Vars”. The next time you load that slot, the functions and shading reappear Which is the point..

Q: Do I need to turn off the “Plot1” feature for every new inequality?
A: Not necessarily, but it’s safer to keep it off while you’re shading. Plot1 draws discrete points, which can mask the shaded region and make it look spotty.

That’s it. Also, with a few button presses and a sensible test point, the TI‑84 Plus becomes a decent visual aid for inequalities. Next time the teacher asks you to shade the feasible region of a linear program, you’ll have the calculator doing the heavy lifting while you focus on interpreting the answer. Happy graphing!