Solving equations and inequalities by graphing

See also: graphing calculator and the TI-84 quick reference

Solving equations on the calculator will be frequently assessed in both Paper 2 and Paper 3 (HL).

An equation, such as $2x + \sin(3x - 5) = \e^{\frac x2}$, can be solved by finding the intersections of

or by finding the zeros of

While equivalent, the zoom is easier to set for zeros, as the $y$-zoom only have to be near the $x$-axis.

Contents

• Solver vs Graphing
• Integer inputs
• System of equations
• Inequalities
• Practice time!

Solver vs Graphing

When you know the number of solutions and it is either one or two, then the numeric solver is ideal. However when the number of solutions is unknown, graphing is superior when you know how to speed it up and use an appropriate zoom. The main advantages of graphing is that you can see all the intersections on the graph, meaning you are less likely to miss solutions. Intersections are best convert to zeros of the difference of two functions, for easier zooms.

In either case, first store the expression in a function, so you can always switch into graphing if needed. Details are discussed in the calculator links above.

Integer inputs

Equations solved over integers, eg involving the binomial coefficient, are best solved using a table of values. It’s probably easiest to find zero of $f(x) - g(x) = 0$.

System of equations

All approved graphing calculators should have a dedicated, built-in app or mode for solving systems of two or three-variables linear equations. They can automatically solve for cases of zero, one, or infinite number of solutions. SL students only have to solve two-variable linear systems.

HL students may, on very rare occasions, be asked to solve a system of non-linear equations. You have to get them either to a single equation with one variable, or a linear system of 2 equations.

Strategies include:

• (substitution) solving for a variable or expression and substitute that into the other equation,
• (elimination) adding up scalar multiples of each equation so a variable cancel out,
• dividing one equation by another, and
• making a substitution and convert the system into a linear system; solve the system; then solve for your original variables by solving the substitution(s).

Inequalities

See also: interpretation of graphs

Note: SL students only need to be able to solve linear and quadratic inequalities.

Solve the equation before solving the inequality. Then on the graph, identify intervals where $f$ is above $g$ (for $f(x) > g(x)$) or below $g$ (for $f(x) < g(x)$). Consider all intervals.

Include the intersection $x$-values when the inequality is nonstrict, ie $\geq$ or $\leq$. Otherwise for strict inequalities exclude them

The possible boundaries for the intervals are the intersections, $+\infty$, $=\infty$ and discontinuities in the graph.

Practice time!

• Solve $2x+\sin(3x - 5) = \e^{\frac x2}$ Ans: $0.0181, 0.544, 1.38, 4.56$
• Solve $-3x^2 - \sqrt{2}x + 6 < \e x + 1$ Ans: $\left]-\infty, -2.15\right[\cup\left]0.774, \infty\right[$

Note SL students only will encounter linear and quadratic inequalities.

• Solve $2x+\sin(3x - 5) \geq \e^{\frac x2}$ Ans: $\left[0.0181, 0.544\right]\cup\left[1.38, 4.56\right]$

This following HL question is very hard and is unlikely to appear on exams.

• Solve the system:

Answer: $(x, y) = (0.190, 3.83)$