Solving Equations and Inequalities

Expert reviewed • 21 July 2024 • 6 minute read

HSC Maths Advanced Syllabus

use graphical methods with supporting algebraic working to solve a variety of practical problems involving any of the functions within the scope of this syllabus, in both real-life and abstract contexts
- determine the number of solutions of an equation by considering appropriate graphs
- solve linear and quadratic inequalities by sketching appropriate graphs

Note:

Video coming soon!

Solving Inequalities Algebraically

To solve inequalities algebraically we need to understand some basic rules:

The absolute value of a number/function will alter the sign of the inequality.

|x|=x \qquad for\quad x\geq0\\|x|=-x\qquad for\quad x<0

In the process of applying multiplication or division to both sides of an inequality using a negative number, the inequality symbol is reversed.
Whenever there is a denominator in the inequality, it can be removed by multiplying both sides by its square.

For example: To solve the following we must apply the above rule.
$\frac{x-5}{4}\geq 2 \\\frac{x-5}{4} \times 4^2 \geq2\times 4^2\\4x-20\geq 32\\4x\geq52\\x\geq13$

Solving Equations and Inequalities Graphically

Solving equations and inequalities graphically involves finding the points on a graph where certain conditions are met. Graphically solving equations and inequalities, is generally done by graphing both right and left hand sides of an equation/inequality on a singular Cartesian plane, and determining the solution visually.

To graphically solve the functions $f(x)$ and $g(x)$ :

$f(x)=g(x)$ : the solution is determined by the points of intersection between the two different graphs.
$f(x)> g(x)$ : the solution is determined by the finding where values of graph $f(x)$ , lie above that of $g(x)$ .
$f(x) <g(x)$ : the solution is determined by the points of the graph produced by $f(x)$ , which sit below that of the curve produced by $g(x)$ .
If an inequality uses either of the following two symbols $\leq or \geq$ , the corresponding rules above are applied. However, the points at which the graphs intercept is also considered a solution.

Practice Question 1

Determine the following solutions for the inequality $x^2>x+2$

The first step in determining the solution to this problem graphically, is to construct a graph of both functions on either side of the inequality (this graph should be to scale). This includes determining all points of intersection, and any intercepts.

Knowing this information, we must now determine, which parts of the curve $x^2$ sit above the curve $x+2$

$\therefore$ Analysing the graph we can see that the solutions to the problem are $x<-1$ and $x>2$

Graph Sketching Made Easy

Up Next

Regions of a Plane

Return to Module 2: Graphs and Equations