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Graphing Slope, Undefined Slope, Flat Slope Area of a Triangle Area of Isosceles Triangle Order of Operations Area of a Sector Names of Shapes Adding/Subtracting FractionsAs well as solving quadratic equations, solving quadratic inequalities is also possible.

Quadratic Inequalities are very similar to Quadratic Equations.

Thus solving inequalities is similar to solving equations.

There are  4 inequalities symbols that can be encountered. Often involving **0**, but
other numbers can be used.

< LESS THAN

> GREATER THAN

≤ LESS THAN OR EQUAL TO

≥ GREATER THAN OR EQUAL TO

When solving quadratic equations, we’re trying to find the values where the equation is equal to
zero, which is where on a graph the equations curve touches the ** x**-axis.

With the case of solving quadratic inequalities however, we’re generally trying to find the values of
** x** where the curve of a specific equation, is either
above or below the

Which is usually over a certain interval of points, as opposed to a specific single point.

The image below shows examples of such intervals in blue.

The approach to solving a quadratic inequality, is to first set the inequality as

an equation equal to  0.

So that we can find the values where the equation equals zero, if such values
exist.

As in between and outside of those values where the equation equals zero, the equation will produce
other values, which are either greater than zero ( **> 0** ), or less than zero ( **<
0** ).

Plugging a number from between or outside of the zero values into the equation, can tell you what
the curve is doing in that period.

Solve

Take the expression **x**^{2} − **1**, while also observing the curve.

Treat an inequality like this firstly as an equation. Set equal to zero, and solve.

Confirms that curve crosses * x*-axis at

Next step is to pick a value between the two solutions, and check if it gives a positive or negative result when used in the equation.

Here

So:

There are though, several methods of notation for solving quadratic inequalities that could be used.

If one were to say

that is usually accepted as a sufficient answer also, and shows understanding.

Further Examples

Solve

Set

=> (

For a value between -1  and 4, can try 0.

So:

**(1.3)**

Solve **2 x**

This solving quadratic inequalities example is slightly different.

The underline "__<__" changes the question somewhat, so that we're looking to find out where
the curve of the equation is either below **or** also on the * x*-axis.

So solving where is the equation less than

The approach to solve though, is exactly the same.

=> (

For a value between -1  and 2, can try 0.

So:

**(1.4)**

Solve **-x**^{2} +
**9** ≥ **0**.

Similar to (1.3), this example is asking where the curve will be either above, or on the x-axis.

So where is the equation greater than or equal to zero.

__Solution__

**-x**^{2} + **9** = **0**

=> **-x**^{2} =
**-9** , * x* =

For a value between -3  and 3, can try 0.

So:

The solution is between

Below is the graph of

**(1.5)**

Solve **x**^{2} − **4 x** +

=> (

In this solving quadratic inequalities example, there is just  1 solution, only  1
point where the curve touches the * x*-axis.

The answer is * x* =

As we were asked to solve where the equation is either greater than, or equal to zero.

As can be seen from the graph, at every other value on the

So:

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