Using Graphs to Solve Equations and Inequalities Revision Notes for HSC SSCE Mathematics Advanced

Using Graphs to Solve Equations and Inequalities

Introduction to graphical methods

Graphical methods provide a powerful way to solve equations that are difficult or impossible to solve algebraically. The main advantages of using graphs are:

You can quickly see how many solutions exist by counting intersection points
You can identify if solutions exist at all
You can find approximate values for solutions
The visual representation makes the mathematical situation clear
Once you understand the situation graphically, you can often find exact solutions algebraically

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The advantage of graphical approaches is that once the graphs are drawn, it is usually obvious from the picture how many solutions there are and roughly where they are located. This visual understanding often guides us toward finding exact algebraic solutions.

Constructing two functions from a given equation

To solve an equation graphically, you create two separate functions from the equation.

The basic method

For an equation like $2^x = x + 2$ :

Take the left-hand side (LHS) and create the function $y = 2^x$
Take the right-hand side (RHS) and create the function $y = x + 2$
Graph both functions on the same set of axes
Find where the graphs intersect

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Sometimes it helps to rearrange the equation before creating your two functions. Choose a rearrangement that makes both functions easy to sketch.

Example: Solving $2^x = x + 2$

Let's examine this equation that cannot be solved algebraically.

When we graph $y = 2^x$ and $y = x + 2$ together, we can see:

The graphs intersect twice, so there are two solutions
One solution is exactly $x = 2$ , because $2^2 = 4 = 2 + 2$ , so the point $(2, 4)$ lies on both graphs
The other solution is just to the right of $x = -2$ , approximately $x \approx -1.7$

Refining approximate solutions

When you find an approximate solution from the graph, you can make it more accurate using:

Careful plotting on graph paper with precise scales
Trial and error on a calculator, testing values near your estimate
Graphing software that can calculate intersection points

Counting the number of solutions

Often you only need to know how many solutions exist and roughly where they are located, rather than exact values.

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Worked Example: Investigating $9 - x^2 = \frac{1}{x}$

Let's determine how many solutions this equation has.

Step 1: Graph both functions

$y = 9 - x^2$ (a parabola opening downward with maximum at $y = 9$ )
$y = \frac{1}{x}$ (a hyperbola)

Step 2: Count the intersection points

The two graphs intersect at three points, so the equation has three solutions:

One solution just to the left of $x = -3$
One solution just to the right of $x = 0$
One solution just to the left of $x = 3$

Key method: Graphical solution of an equation

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To solve an equation graphically:

Sketch the graphs of $y = \text{LHS}$ $y = LHS$ and $y = \text{RHS}$ $y = RHS$ on one pair of axes
- Consider whether rearranging the equation first would be helpful
The solutions are the x-coordinates of any points of intersection
The graph shows you:
- How many solutions exist
- Approximate locations of the solutions

This method gives you both quantitative information (how many solutions) and qualitative information (where they are).

Solving inequations using graphs

Graphical methods are equally useful for inequations (inequalities). The key is to compare which function lies above the other.

Example: Solving $2^x < x + 2$

Using our earlier graph where $y = 2^x$ and $y = x + 2$ intersect at approximately $x = -1.7$ and $x = 2$ :

The inequality $2^x < x + 2$ asks: where is the curve $y = 2^x$ below the line $y = x + 2$ ?
From the graph, this occurs between the two intersection points
Solution: $-1.7 < x < 2$ (approximately)

Key method: Graphical solution of an inequation

chatImportant

To solve an inequation graphically:

Sketch the graphs of $y = \text{LHS}$ and $y = \text{RHS}$ on one pair of axes
Examine which curve lies above or below the other at different values of $x$

For inequalities:

$\text{LHS} > \text{RHS}$ : Find where the LHS curve is above the RHS curve
$\text{LHS} < \text{RHS}$ : Find where the LHS curve is below the RHS curve
$\text{LHS} \geq \text{RHS}$ or $\text{LHS} \leq \text{RHS}$ : Include the intersection points in your solution

Absolute value equations and inequations

Absolute value problems benefit greatly from graphical solutions because the sketch makes the situation clear.

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Worked Example: Solving $|2x - 5|$ equations and inequations

Let's work through a complete example involving the absolute value function $y = |2x - 5|$ .

Part a: Draw the graph of $y = |2x - 5|$

To sketch this absolute value function:

Find the x-intercept: Set $2x - 5 = 0$

$x = 2\frac{1}{2}$

Find the y-intercept: Set $x = 0$

$y = |0 - 5| = 5$

The graph forms a V-shape with vertex at $x = 2\frac{1}{2}$ .

Part b: Write equations for both branches

The absolute value function has two linear pieces:

For $x \geq 2\frac{1}{2}$ : $y = 2x - 5$ (right-hand branch)
For $x < 2\frac{1}{2}$ : $y = -2x + 5$ (left-hand branch)

Part c: Draw $y = x + 2$ on the same diagram

Part d: Find the intersection points P and Q

We solve algebraically for each intersection:

Finding P (right branch):

x + 2 = 2x - 5 2 = x - 5 x = 7

So $P = (7, 9)$

Finding Q (left branch):

x + 2 = -2x + 5 3x = 3 x = 1

So $Q = (1, 3)$

Part e: Solve $|2x - 5| = x + 2$

From the graph, the two functions intersect at points P and Q.

Solution: $x = 1$ or $x = 7$

Part f: Solve $|2x - 5| \geq x + 2$

We need to find where the V-shaped graph $y = |2x - 5|$ is on or above the line $y = x + 2$ .

From the graph, this occurs:

To the left of Q (including Q): $x \leq 1$
To the right of P (including P): $x \geq 7$

Solution: $x \leq 1$ or $x \geq 7$

infoNote

When working with absolute value inequations, pay careful attention to whether the inequality includes the boundary points (≤ or ≥) or not (< or >). This determines whether you include the x-coordinates of the intersection points in your solution.

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Key Points to Remember:

Graphical solutions show the big picture: You can immediately see how many solutions exist and approximately where they are
Solutions are x-coordinates of intersections: Where $y = \text{LHS}$ and $y = \text{RHS}$ cross, the x-value is a solution
For inequations, compare heights: Determine which function is above or below the other across different regions
Absolute value creates V-shapes: Split the absolute value function into two linear pieces at the vertex
Verify algebraically when possible: Once the graph reveals the structure, find exact solutions using algebra

Using Graphs to Solve Equations and Inequalities (HSC SSCE Mathematics Advanced): Revision Notes