Solving Equations and Inequalities

We can solve equations of the form $f(x) = g(x)$ or inequalities like $f(x) < g(x)$ using two main approaches: graphical (using the intersections of curves) or algebraic (using calculation).

Graphical Method

Method Solving Graphically

To solve equations and inequalities using the graphs of $f$ and $g$:

For $f(x) = g(x)$: Locate the intersection points of the two curves. The solutions are the x-coordinates (abscissas) of these points.
For $f(x) < g(x)$: Identify the intervals of $x$ where the curve of $f$ is below the curve of $g$.
For $f(x) > g(x)$: Identify the intervals of $x$ where the curve of $f$ is above the curve of $g$.

Example

The functions $f$ and $g$ are represented below. Solve $f(x) = g(x)$ and $f(x) \leq g(x)$.

Answer

Equation $f(x) = g(x)$: The curves intersect at $x = -1$ and $x = 2$. The solution set is $\boldsymbol{S = \{-1 ; 2\}}$.
Inequality $f(x) \leq g(x)$: The curve of $f$ is below or touching the curve of $g$ between the two intersection points. The solution set is the interval $\boldsymbol{S = [-1 ; 2]}$.

Algebraic Method

Method Solving Algebraically

To solve $f(x) = g(x)$ algebraically:

Set the expressions for $f(x)$ and $g(x)$ equal to each other.
Move all terms to one side of the equation to obtain an expression equal to zero (e.g., $f(x) - g(x) = 0$).
Factorize or simplify to find the exact values of $x$.

Example

Let $f(x) = 3x + 1$ and $g(x) = x + 7$. Solve $f(x) = g(x)$.

Answer

$$\begin{aligned} f(x) &= g(x) \\ 3x + 1 &= x + 7 \\ 3x - x &= 7 - 1 \\ 2x &= 6 \\ x &= 3\end{aligned}$$The solution is $\boldsymbol{x = 3}$.