Solving inequalities using graphs

3 Solving inequalities using graphs

Graphs can be used to help solve inequalities. This approach is particularly useful if the inequality is not linear as, in these cases solving the inequalities algebraically can often be very tricky. Graphics calculators or software can save a lot of time and effort here.

Example 38

Solve graphically the inequality $5 x + 2 < 0$ .

Solution

Figure 12 :

{ Graph of $y=5x+2$.}

We consider the function $y = 5 x + 2$ whose graph is shown in Figure 12. The values of $x$ which make $5 x + 2$ negative are those for which $y$ is negative. We see directly from the graph that $y$ is negative when $x < - \frac{2}{5}$ .

Example 39

Find the range of values of $x$ for which $x^{2} - x - 6 < 0$ .

Solution

We consider the graph of $y = x^{2} - x - 6$ which is shown in Figure 13.

Figure 13 :

${ Graph of $y=x^2-x-6$}$

Note that the graph crosses the $x$ axis when $x = - 2$ and when $x = 3$ , and $x^{2} - x - 6$ will be negative when $y$ is negative. Directly from the graph we see that $y$ is negative when $- 2 < x < 3$ .

Task!

Find the range of values of $x$ for which $x^{2} - x - 6 > 0$ .

The graph of $y = x^{2} - x - 6$ has been drawn in Figure 13. We require

$y = x^{2} - x - 6$ to be positive.

Use the graph to solve the problem:

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

Example 40

By plotting a graph of $y = 20 x^{4} - 4 x^{3} - 143 x^{2} + 46 x + 165$ find the range of values of $x$ for which

$20 x^{4} - 4 x^{3} - 143 x^{2} + 46 x + 165 < 0$

Solution

A software package has been used to plot the graph which is shown in Figure 14. We see that $y$ is negative when $- 2.5 < x < - 1$ and is also negative when $1.5 < x < 2.2$ .

Figure 14 :

${ Graph of $y=20x^4-4x^3-143x^2+46x+165$}$

Exercises

In questions 1-5 solve the given inequality graphically:

$3 x + 1 < 0$
$2 x - 7 < 0$
$6 x + 9 > 0$ ,
$5 x - 3 > 0$
$x^{2} - x - 6 < 0$

$x < - 1 ∕ 3$
$x < 7 ∕ 2$ ,
$x > - 3 ∕ 2$
$x > 3 ∕ 5$
$- 2 < x < 3$

3 Solving inequalities using graphs

Answer

Answer