Inequalities on graphs

In a nutshell

Graphical inequalities can be interpreted to identify solutions to inequality problems. 

Solutions to inequalities

When being asked to solve an inequality of the form $$f(x)<g(x)$$​ or $$f(x)>g(x)$$​, it may be helpful to sketch the graphs of $$y=f(x)$$ and $$y=g(x)$$. The inequality can be solved by looking at their relative positions on the graph.

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Example

$$L_{1}$$ has the equation $$y= x^2+2x$$.

$$L_{2}$$ has the equation $$y=-3x+6$$.

The diagram shows a sketch of $$L_{1}$$ and $$L_{2}$$.

i) Find the coordinates of the points of intersection, $$P_{1}$$ and $$P_2$$.

ii) Find the solution to the inequality $$x^2+2x \lt -3x+6$$.  

Part i):

The points of intersection can be found where $$L_1 = L_2$$:

$$x^2+2x = -3x+6$$

Rearrange and solve to find $$x$$:

$$x^2+5x-6 = 0\\ x = \dfrac{-5 \pm \sqrt{5^2 - 4(1)((-6)}}{2(1)}$$

$$x=1,\ x=-6$$​​​​

Substitute these values for $$x$$ into one of the equations to find coordinates of $$P_1$$ and $$P_2$$:

$$L_1: y=-3x+6$$

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When $$x=1$$:

$$y = -3(1)+6 = 3$$​​

When $$x=-6$$:

$$y=-3(-6)+6 = 24$$​​

Therefore:

​The points of intersection are $$\underline{P_1:(6,24)}$$ and $$\underline{P_2:(1,3)}$$.

Part ii):

The solution to the inequality is when the graph of $$y=-3x+6$$ is above the graph of $$y=x^2+2x$$. 

Therefore:

$$\underline{\{x:-6\lt x\lt 1\}}$$​​