Quadratic inequalities

Quadratic inequalities are inequalities that involve a quadratic expression. For example: 5x^2 - 3x + 2 > 0.

They’re a little trickier to solve than regular quadratic equations, because we need to find the range of values that satisfy the inequality, rather than just specific values.

The basic steps

A summary of the steps to solve quadratic inequalities is:

Rearrange the inequality so that one side is zero (e.g., ax^2 + bx + c > 0)
Solve the equation as if it were a normal quadratic equation (ax^2 + bx + c = 0)
Sketch the graph of that equation
Using the inequality sign, determine whether we are looking for the parts above the x-axis (for > or ≥) or below the x-axis (for < or ≤).
Find the range of x values from the graph that would produce this.
Write that as an inequality or in interval notation.

Examples

Example: Solve the inequality x^2 - 4x + 3 < 0.

Treat this as a normal quadratic equation first:
- x^2 - 4x + 3 = 0
- (x - 1)(x - 3) = 0
- So, x = 1 or x = 3.
Sketch the graph of the inequality x^2 - 4x + 3 < 0:
- [I will draw the graph later when I have time]
Since the inequality is ‘<’, we are looking for the parts of the graph that are below the x-axis.
Between the roots, the y-values are negative (below the x-axis), and so we are looking for the range between the roots of 3 and 1.
Finally, we write that as an inequality:
- 1 < x < 3

Set notation

TODO: rewrite this subheading. For completeness, some parts of this section are AI generated and will likely be incorrect or poorly explained - this will soon be fixed.

When writing the solution to a quadratic inequality, we can use set notation to clearly express the range of values that satisfy the inequality.

For example, if we have solved the inequality x^2 - 4x + 3 < 0 and found that the solution is 1 < x < 3, we can express this in set notation as:

\{ x \in \mathbb{R} \mid 1 < x < 3 \} This reads as “the set of all real numbers x such that x is greater than 1 and less than 3”.

Interval notation

TODO: rewrite this subheading. For completeness, some parts of this section are AI generated and will likely be incorrect or poorly explained - this will soon be fixed.

Another way to express the solution to a quadratic inequality is through interval notation.

Using the same example of the inequality x^2 - 4x + 3 < 0 with the solution 1 < x < 3, we can express this in interval notation as:

(1, 3) This indicates that x can take any value between 1 and 3, but does not include the endpoints 1 and 3 themselves (which is why we use parentheses instead of brackets).

If the inequality were inclusive (e.g., x^2 - 4x + 3 \leq 0), the solution would include the endpoints, and we would write it in interval notation as:

[1, 3] This indicates that x can take any value between 1 and 3, including the endpoints 1 and 3 (which is why we use brackets).

Using set notation or interval notation helps to clearly communicate the solution to quadratic inequalities in a concise manner, allowing for elaborate sharing of mathematical ideas.

flashcards

Question	Answer
What is a quadratic inequality?	An inequality involving a quadratic expression, e.g. 5x^2 - 3x + 2 > 0.
What is the key difference between solving a quadratic equation and a quadratic inequality?	For an inequality, you find a range of values that satisfy it, rather than just specific values.
What is the first step when solving a quadratic inequality?	Rearrange the inequality so that one side is zero, e.g., ax^2 + bx + c > 0.
After rearranging a quadratic inequality, what do you do with the corresponding equation?	Solve the equation as if it were a normal quadratic equation (ax^2 + bx + c = 0).
After solving the quadratic equation, what is the purpose of sketching its graph?	To determine which parts of the graph are above or below the x-axis based on the inequality sign.
For a quadratic inequality with ‘>’ or ‘≥’, which part of the graph do you look for?	The parts above the x-axis.
For a quadratic inequality with ‘<’ or ‘≤’, which part of the graph do you look for?	The parts below the x-axis.
Solve the inequality x^2 - 4x + 3 < 0.	The roots are x = 1 and x = 3. The solution is 1 < x < 3.
Express the solution 1 < x < 3 in set notation.	\{ x \in \mathbb{R} \mid 1 < x < 3 \}
Express the solution 1 < x < 3 in interval notation.	(1, 3)
How do you express the inclusive solution 1 \le x \le 3 in interval notation?	[1, 3]
Why are parentheses used in interval notation instead of brackets?	Parentheses indicate the endpoints are not included in the solution.
Why are brackets used in interval notation?	Brackets indicate the endpoints are included in the solution.