E2.6 - Solving inequalities with graphs (Extended)

Equality boundary

For $f(x)>g(x)$, the equality $f(x)=g(x)$ identifies possible boundaries between intervals where one graph is higher. Locate these using intersections, or zeros of $f(x)-g(x)$. Any input where an expression is undefined must also be considered separately.

Worked example 1: two equality points

Question. Find the equality boundaries needed to solve $x^2-x-6>0$.

Plan. Find zeros of the expression on the left.

1. Enter $Y_1=X^2-X-6$, using $-4\le x\le5$, $-8\le y\le12$.
2. Use the zero feature between $-3$ and $-1$, then between 2 and 4.
3. The zeros are $-2$ and 3, so they split the line into three intervals.

Answer. The equality boundaries are $x=-2$ and $x=3$.

Check. $(-2)^2-(-2)-6=0$ and $3^2-3-6=0$. These are boundary inputs, not yet the full inequality solution.

Worked example 2: an unfamiliar increasing function

Question. Find the equality boundary for $2^x+x<6$ on $0\le x\le4$.

Plan. Intersect the function graph with the horizontal line of height 6.

1. Enter $Y_1=2^X+X$, $Y_2=6$, using $0\le x\le4$, $0\le y\le21$.
2. The intersection feature between 1 and 3 gives $(2,6)$.
3. The graph rises throughout this interval, so there is only one equality point.

Answer. The equality boundary is $x=2$.

Check. $2^2+2=6$. Outputs at 1 and 3 are 3 and 11, on opposite sides of 6. Both $2^x$ and $x$ increase, so their sum cannot return to 6 later in the interval.

Watch out

The answer $x=-2$ or $x=3$ describes the equality in the first example. The strict inequality excludes those inputs and needs intervals selected from either side.

Strict and inclusive endpoints

The signs $<$ and $>$ exclude equality boundaries. The signs $\le$ and $\ge$ include them, provided the original expressions are defined there. On a number line use open circles for excluded endpoints and closed circles for included endpoints.

Worked example 3: a strict inequality

Question. Solve $x^2-x-6>0$ graphically.

Plan. Use the zeros $-2$ and 3, then select where the graph is above zero.

Test one input in each interval:

The upward parabola stays above the axis outside its two crossings. The strict sign excludes the zeros.

Answer. $x<-2$ or $x>3$.

Check. The test inputs $-3$ and 4 satisfy the original inequality; the interior input 0 does not. At either boundary the output is zero, which fails $>0$.