Graphing inequalities on the TI-84 is a skill that takes about five minutes to learn and saves considerable time on homework and exams. From simple linear inequalities to complex systems involving quadratic boundaries, the TI-84 can shade solution regions automatically, find boundary intersections precisely, and verify answers that students have worked out by hand.
Introduction
Graphing inequalities on the TI-84 is a skill that takes about five minutes to learn and saves considerable time on homework and exams. From simple linear inequalities to complex systems involving quadratic boundaries, the TI-84 can shade solution regions automatically, find boundary intersections precisely, and verify answers that students have worked out by hand.
The graphical approach to inequalities has an important pedagogical advantage over the algebraic approach: it makes the solution set visible and intuitive. When students can see the shaded region that represents all points satisfying the inequality, the concept of a solution set as a region of the coordinate plane becomes concrete rather than abstract. This visual understanding transfers directly to higher-level topics like linear programming and multivariable calculus.
This guide covers linear inequalities using the TI-84's shading feature, systems of inequalities with overlapping shaded regions, quadratic inequalities using the zero-finding approach, and finding intersection points of boundary curves — an essential technique for linear programming problems.
Linear Inequalities: The Shading Method
The TI-84 can shade regions above or below a curve using a special plot style in the Y= editor. For the inequality y less than 2x + 3, press Y= and enter 2X+3 as your boundary function in Y1. Move the cursor all the way to the left past the Y1 label to the small line style icon. Press ENTER repeatedly to cycle through the available plot styles. Stop at the shade-below icon (which looks like a downward triangle or wedge) for y less than or y less than or equal to inequalities. For y greater than inequalities, select the shade-above icon.
Press GRAPH — the calculator draws the boundary line and automatically shades the appropriate region. The shaded area represents all coordinate points that satisfy the inequality. You can use TRACE to verify specific points: if a point lies in the shaded region, its coordinates satisfy the original inequality.
An important limitation: the TI-84 shading does not distinguish between strict inequalities (less than, greater than) and non-strict inequalities (less than or equal to, greater than or equal to). The graph looks the same for both. For exam work, you need to draw the boundary as a dashed line for strict inequalities and a solid line for non-strict inequalities in your written solutions.
Systems of Inequalities
Graphing a system of two or more inequalities works by entering each boundary function with its appropriate shade style. For the system y less than x + 4 AND y greater than negative x + 2, enter x+4 in Y1 with shade-below style and enter -x+2 in Y2 with shade-above style. Press GRAPH — both regions are shaded simultaneously.
The overlapping double-shaded area where both shading patterns coincide represents the solution set of the system — all points that satisfy both inequalities simultaneously. If the shading regions do not overlap anywhere, the system has no solution, meaning no point in the coordinate plane satisfies all inequalities at once. Visually identifying this case is much easier with the TI-84 than through algebraic analysis.
For systems of three or more inequalities, continue adding functions with appropriate shade styles in Y3, Y4, and so on. The feasible region — where all shadings overlap — represents the solution set. For linear programming problems, this feasible region is a polygon whose vertices are the solutions to pairs of boundary line equations. Use the intersect feature to find these vertices precisely.
Quadratic Inequalities
For a quadratic inequality like x squared minus 4 greater than 0, graph Y1 = X^2-4 using ZStandard to get a clean initial view. Find the zeros using 2ND+TRACE → 2:zero: set bounds around each crossing and press ENTER three times. You will find zeros at x = -2 and x = 2. The inequality is satisfied where the parabola lies above the x-axis: when x is less than -2 or x is greater than 2.
The solution in interval notation is negative infinity to negative two union two to positive infinity. Reading this directly from the graph eliminates the sign chart analysis that many algebra textbooks require: you simply identify where the curve is above the x-axis rather than constructing a systematic sign chart.
For more complex quadratic inequalities involving comparisons between two quadratic functions, enter both functions in Y1 and Y2 and compare their graphs directly. The inequality Y1 greater than Y2 holds wherever the Y1 graph lies above the Y2 graph. Find the intersection points to determine the boundary values of the solution intervals.
Finding Intersection Points of Boundary Lines
For a system where you need to find the vertices of the feasible region, the intersect function is essential. For a system with boundaries y = 3x + 1 and y = x squared minus 5, graph both functions and use 2ND+TRACE → 5:intersect. Set a left bound to the left of one intersection, a right bound to its right, and a guess near the crossing. Press ENTER three times to get the precise intersection coordinates.
These intersection points are the vertices of the feasible region and are critical for linear programming optimization problems. The maximum or minimum of an objective function over a feasible polygon region always occurs at one of the vertices. Finding these vertices precisely with the TI-84 is faster and more accurate than solving systems of boundary equations by hand.
For systems with three or more boundary curves, repeat the intersect process for each pair of adjacent boundaries. Label each vertex with its coordinates and then evaluate your objective function at each vertex. The vertex producing the maximum or minimum objective value is your answer. This entire process, which can take many minutes algebraically, takes two to three minutes with the TI-84 using this approach.
Further Reading & Sources
Setting Up Inequality Shading
On the TI-84, inequality graphing works by modifying the shading style for each Y= function. In the Y= editor, scroll left past "Y1=" to the backslash symbol (\). Each press of ENTER cycles through five styles: normal line, thick line, shade above (▲), shade below (▼), and animate. For inequalities, use shade above for ≥ or > and shade below for ≤ or <. The boundary line itself is always solid — the TI-84 cannot draw dashed lines for strict inequalities.
Graphing a Single Linear Inequality
To graph y ≥ 2x − 3: Enter Y1 = 2X − 3. In the Y= editor, scroll left to the Y1 style icon and press ENTER until you see the shade-above triangle (▲). Press GRAPH. The calculator plots the line and shades the region above it. Use ZOOM → 6:ZStandard for a standard −10 to 10 window, or manually set WINDOW values appropriate to your problem.
Systems of Inequalities
For systems with two or more inequalities, enter each as a separate Y function with the appropriate shading direction. The feasible region — where all constraints are satisfied simultaneously — is where the shaded areas overlap. On the TI-84, overlapping shading appears as a slightly different texture or density. For linear programming problems, look for the vertices of the feasible region, which you can find using 2ND → CALC → 5:intersect between pairs of boundary lines.
Finding the Feasible Region Vertices
Linear programming requires identifying corner points of the feasible region. After graphing all boundary lines with appropriate shading, use 2ND → CALC → 5:intersect for each pair of adjacent boundary lines. Move the cursor near the intersection point when prompted. The TI-84 returns the exact coordinates. Evaluate the objective function at each vertex to find the maximum or minimum value.
Quadratic and Nonlinear Inequalities
The TI-84 handles quadratic inequalities the same way as linear ones — set the shading style in Y=, enter the quadratic function, and shade above or below. For y < x² − 4, enter Y1 = X²−4 with shade below. The parabola itself is the boundary. For absolute value inequalities like y ≥ |x+2|, enter Y1 = abs(X+2) (abs is under MATH → NUM → 1:abs) with shade above.
Common Mistakes to Avoid
The most frequent errors when graphing inequalities: (1) Forgetting to change the shading style — the default is a normal line with no shading. (2) Confusing shade-above and shade-below — always think of it as "the solution is the region above/below the boundary line." (3) Having too many Y functions active — turn off unused Y functions by pressing ENTER on the = sign in Y= until it's not highlighted. (4) Misidentifying the feasible region — remember you need ALL conditions satisfied simultaneously, not just one.