For the past week, I have learned how to write solutions for quadratic inequalities and also how to graph them.

Let’s start with an example:

Example 1: Given an equation

First step is take out what is common then factor to be able to find the x-intercepts.

1. Common: 2

2()<0

2. Factor

2(x+1)(x-5) < 0

x-intercepts: -1 and 5

Now that you have the x-intercepts, we can draw these on a number line.

By looking at the equation, we can tell the parabola will be opening up as the coefficient next to is positive. So there is a +, then a – , then a + as the parabola will dip past the x-axis since it has two x-intercepts.

Looking at the inequality symbol, it’s using <. This means they are asking for the numbers that are less than 0, so negative numbers.

 

 

The negative numbers are in the middle so the solution would look like this:

-1 < x < 5

The circles are not coloured because it is not including “or equal to”. This equation is just using less than.

Now let’s look at another example where we are given a graph and we must write an inequality.

Let’s go over some important things:

≤ and ≥ : has a solid line

< and >: has a dash line

Example:

First step is to find the y intercept, which is -1. Next is to find the slope. We can tell by the points that it’s going up by 4 and moves to the right once. So the slope is 4. This linear graph is tilted up so it’s positive.

The equation so far looks like this:

y ☐ 

To find the inequality symbol, we must test a number from the shaded section that is not a point on the line and determine which symbol to choose which will result in a correct answer.

Let’s use (5,0)

Plug in 5 in x and 0 in y:

0 ☐

0 ☐ 4(25)-1

0 ☐ 100-1

0 ☐ 99

To make this true, we must use ≤ because 99 is greater than 0 and the line is solid. 

So the equation will be:

y ≤