1.4. Solving an equation or inequality graphically/numerically

Example questions: solve the equation $\mathrm{e}^x = 5-x^2$ numerically.
Solve the inequality $\mathrm{e}^x \leq 5-x^2$ , leaving your answer correct to 3 decimal places.

Only specific types of equations or inequalities (for example, the linear $2x-5 \leq 3$, the quadratic $ x^2 - 3x -4 > 0$, the rational $\displaystyle \frac{x+2}{x-1} \geq 2$ and a few other cases) admit an algebraic method leading to an exact solution. For everything else, we use a graphical/numerical method with our graphing calculator.

Solution (without comments)

Solution to $\mathrm{e}^x = 5-x^2$: $x=-2.21 \quad \mathrm{or} \quad x=1.24 $.

Solution to $\mathrm{e}^x \leq 5-x^2$: $-2.211 \leq x \leq 1.241 $.

Method 1: Sketch two curves and use the "intersect" solver

Since we have $\mathrm{e}^x$ and $5-x^2$ on separate sides of the equation/inequality, we sketch two curves: $ y = \mathrm{e}^x$ and $y=5-x^2$ using our graphing calculator.

We observe that there are two points of intersection between the two curves. We then use the "intersect" solver in the calculator twice, "guessing" close to one of the answers each time to obtain the coordinates of both points of intersection.

This solves the equation $\mathrm{e}^x = 5-x^2$, giving us the solution $x=-2.21 \quad \mathrm{or} \quad x=1.24 $ correct to 3 significant figures.

To solve the inequality, we need to observe the parts where the curve $y=\mathrm{e}^x$ is below the curve $y=5-x^2$. For this example, it occurs between the two points of intersection, giving our solution $-2.211 \leq x \leq 1.241 $ correct to 3 decimal places.

Method 2: Make one side 0, sketch one curve and use the "zero" solver

Instead of sketching two curves, we can rearrange the equation/inequality to get $\mathrm{e}^x - 5 + x^2 = 0$ or $\mathrm{e}^x - 5 + x^2 \leq 0$. This allows us to only draw one curve. For this method, we use the "zero" solver in our graphing calculator to find the $x$-intercepts. instead of the "intersect" solver.

Hence the solution for the equation is $x=-2.21 \quad \mathrm{or} \quad x=1.24 $ correct to 3 significant figures.

For the inequality, we must take the region that is less than or equal to zero. That will be the part below the $x$-axis, giving us the solution $-2.211 \leq x \leq 1.241 $ correct to 3 decimal places.