One-Variable Quadratic Inequalities

we have two expressions that are equal to each other, and we write the equals sign, =,between them. When we have two expressions that are not equal to one another, we can relate the expressions by the use of an inequality sign. We can have inequalities such as:

𝑥≥4

6≤𝑥

2𝑥−7>5

In each of these inequalities, 𝑥

 has a range of possible solutions. When we have an inequality such as 𝑥≥4

we can say this in words as “𝑥 is greater than or equal to four.” This means that a value of 𝑥 that is four or more will satisfy this inequality. The four inequality symbols we use are:

>: greater than  

≥: greater than or equal to

<:less than

≤ :less than or equal to

 

At the end of this lesson student will be able to

1) Solve inequalities in one variable algebraically

2) Solve inequalities in one variable graphically

We can solve inequalities in a similar process to solving equations, by ensuring that we perform the same mathematical operation to both sides of the inequality. However, as inequalities have a direction, we must be careful to consider which side of the inequality an expression is on. When we multiply or divide by a negative number, we must switch the inequality. For example, if we have−𝑥≤−2,then when dividing by −1 we must switch the inequality to give𝑥≥2.

Let us now look at how to solve an inequality and represent the answer as an interval. Before doing so, we need to recap some notation. If we consider the interval of numbers from 0 to 10, which includes 0 but not 10, we could represent this using inequalities as0≤𝑥<10.

The strict inequality on the right tells us that the 10 is not included in the inequality, and the nonstrict inequality on the left tells us that the 0 is included. Another way of writing this interval would be [0,10[.

Here, the closed square bracket tells us that the 0 is included, and the open square bracket tells us that the 10 is not included. It is also worth recapping here that the symbol for infinity is ∞.This is often used to represent intervals that are greater than or less than a single number. For example,𝑥>3in interval notation would be ]3,∞[.

In this explainer, we will be focusing on quadratic inequalities. This is in contrast to linear inequalities, which look something similar to the following:−2𝑥+3≤5.

Recall that the procedure to solve inequalities of this form is quite straightforward. The first thing we want to do is rearrange the inequality so all of the 𝑥-terms are on one side and all the constant terms are on the other. We do this by subtracting 3 from both sides:−2𝑥≤2.

Then, to get this in terms of just 𝑥,we divide each side by −2,remembering that when we divide an inequality by a negative number, we need to switch the inequality sign. This gives us𝑥≥−1.

So, 𝑥 is all numbers greater than or equal to −1.This can also be expressed as an interval, as [−1,∞[.

In the same way that we have distinct equations such as linear and quadratic equations.

https://youtu.be/pY8jo8b4XJ4

solve ( x – 3)( x + 2) > 0.( 5min)(10marks)

Solve ( x – 3)( x + 2) = 0. By the zero product property, equation

Make the boundary points. Here, the boundary points are open circles because the original inequality does not include equality 

Select points from the different regions created

equation

See whether the test points satisfy the original inequality.

equation

Since x = –3 satisfies the original inequality, the region x < –2 is part of the solution. Since x = 0 does not satisfy the original inequality, the region –2 < x < 3 is not part of the solution. Since x = 4 satisfies the original inequality, the region x > 3 is part of the solution.

Represent the solution in graphic form and in solution set form.

The solution set form is { x| x < –2 or x > 3}.(3min)(5marks)

Solve 𝑥−2𝑥<0 graphically.

•    Aℝ−]0,2[

•    B{0,2}

•    C[0,2]

•    D]0,2[

•    Eℝ−[0,2]

final Degree (out of 15)

How To: Solving a Quadratic Inequality Algebraically

1.    Rearrange the inequality so that we have all the terms of the expression, defined as 𝑓(𝑥),on one side, with an inequality relating this to zero. For example, 𝑓(𝑥)≤0 or 𝑓(𝑥)>0.

2.    Solve 𝑓(𝑥)=0 by factoring, or otherwise, to find the solutions to the equation.

3.    Select test points for each interval so that there are values less than, between, and greater than the solutions of the equation. We can also use a sign chart to identify the intervals that will be positive or negative.

4.    Identify the intervals that satisfy the inequality.

Quadratic equation in one variable page9

●methods of solving quadratic equation page11

●Examples page 14