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.