Solving Equations and Inequalities Graphically and Numerically

Every equation $f(x)=g(x)$ can be rewritten with everything on one side. Define $h(x)=f(x)-g(x)$. Then $f(x)=g(x)$ exactly when $h(x)=0$, so the solutions of the equation are the zeros of $h$ — the $x$-intercepts of a single graph. This one move pays off three ways. You only have to look at one curve and find where it meets the $x$-axis, which is easier to read than where two curves cross. The sign of $h$ tells you which side is bigger: $h(x)>0$ means $f(x)>g(x)$, which is exactly what you need for inequalities. And a change of sign in $h$ between two inputs is a numerical signal that a solution lies between them, which is the basis of the bisection method.

With a graphing calculator or graphing software you can either graph $y=f(x)$ and $y=g(x)$ and use the intersect feature, or graph $y=h(x)$ and use the zero feature. The danger is the window. Graph $2^x$ and $x^2$ for $0\le x\le 5$ and you will see the crossings at $x=2$ and $x=4$ and miss the third one entirely, near $x\approx -0.767$, because it is off the screen. Before trusting a graph, think about end behaviour: as $x\to -\infty$, $2^x\to 0$ while $x^2$ grows, so $h(x)=2^x-x^2$ is negative far to the left; at $x=0$ it is positive. A sign change between them guarantees another root. Check exact-looking answers by substitution — $2^4=16=4^2$ confirms $x=4$ — and report the others to the accuracy asked for, in radians for any trig equation.

If $h$ is continuous on $[a,b]$ — its graph has no breaks — and $h(a)$ and $h(b)$ have opposite signs, the graph must cross the axis somewhere between them. Bisection turns that fact into an algorithm. Evaluate $h$ at the midpoint $m=\frac{a+b}{2}$. If $h(m)$ has the same sign as $h(a)$, the root is in $[m,b]$; otherwise it is in $[a,m]$. Keep the half that still has a sign change and repeat. Every step halves the interval, so after $n$ steps an interval of width $w$ has shrunk to $\frac{w}{2^n}$: ten halvings take a width of 1 below $0.001$. You stop when the required accuracy is guaranteed — for example, when both endpoints round to the same value at the stated number of decimal places. Bisection is slow compared with a calculator's built-in solver, but it never fails on a continuous function with a sign change, and it is how you can be sure of every digit you report.

Intermediate Value Theorem - Video 4 - The Bisection Method

To solve $f(x)>g(x)$, find the solutions of $f(x)=g(x)$ first; they are the only places where the inequality can switch from true to false. Then read the sign of $h$ on each interval between them, from the graph or by testing one point. For $2^x>x^2$, the sign row above gives $-0.767<x<2$ or $x>4$ (approximately). Endpoints are included for $\ge$ and excluded for $>$. The same reasoning answers applied questions. Suppose Town A has 12 000 people growing by 3 percent per year, $A(t)=12000{(1.03)}^t$, and Town B has 18 000 people growing by 200 per year, $B(t)=18000+200t$. The equation $A(t)=B(t)$ mixes an exponential with a line, so you graph $h(t)=A(t)-B(t)$: it is negative at $t=20$ and positive at $t=25$, and technology places the zero at $t\approx 20.7$ years. So Town A is larger for $t>20.7$ — and because exponential growth eventually beats linear growth, it stays larger.

9.6 Techniques for Solving Equations and Inequalities