Inequalities and Modulus Functions

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Why This Matters

This lesson covers the fundamental concepts of inequalities and modulus functions, essential tools for solving a wide range of mathematical problems in A Level Pure Mathematics. We will explore various methods for solving linear and quadratic inequalities, as well as equations and inequalities involving the modulus function, emphasizing graphical interpretations and algebraic techniques.

Key Words to Know

Inequality — A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥.

Quadratic Inequality — An inequality involving a quadratic expression, typically solved by finding roots and analyzing the parabola's shape.

Modulus Function (|x|) — Defines the absolute value of a number, representing its distance from zero on the number line.

Critical Values — The values of x where an expression equals zero or is undefined, used to define intervals for solving inequalities.

Sign Diagram (Number Line) — A visual tool used to determine the sign of an expression over different intervals.

Absolute Value Equation — An equation involving the modulus function, often leading to two separate linear equations.

Absolute Value Inequality — An inequality involving the modulus function, solved by considering cases or squaring both sides.

Solving Linear Inequalities

Linear inequalities are solved much like linear equations, with one crucial difference: multiplying or dividing by a negative number reverses the inequality sign.

Steps for solving linear inequalities:

Isolate the variable on one side of the inequality.
Perform operations (addition, subtraction, multiplication, division) on both sides.
Remember to reverse the inequality sign if multiplying or dividing by a negative number.

Example: Solve 3x - 5 < 7

Add 5 to both sides: 3x < 12
Divide by 3: x < 4

Example: Solve -2x + 1 ≥ 9

Subtract 1 from both sides: -2x ≥ 8
Divide by -2 (and reverse the sign): x ≤ -4

Solutions are often expressed using interval notation or set notation. For example, x < 4 can be written as (-inf, 4).

Solving Quadratic Inequalities

Quadratic inequalities are typically solved by finding the roots of the corresponding quadratic equation and then analyzing the sign of the quadratic expression.

Steps for solving quadratic inequalities:

Rearrange the inequality so that one side is zero (e.g., ax^2 + bx + c > 0).
Find the roots of the corresponding quadratic equation (ax^2 + bx + c = 0). These are your critical values.
Sketch the parabola or use a sign diagram (number line) to determine the intervals where the inequality holds true.
- If a > 0, the parabola opens upwards.
- If a < 0, the parabola opens downwards.
Write down the solution in interval or set notation.

Example: Solve x^2 - x - 6 > 0

Roots of x^2 - x - 6 = 0 are (x-3)(x+2) = 0, so x = 3 and x = -2.
This is an upward-opening parabola. The expression is positive when x < -2 or x > 3.
Solution: x < -2 or x > 3 (or (-inf, -2) U (3, inf)).

Understanding the Modulus Function

The modulus function, denoted as |x|, gives the absolute value of x. Geometrically, it represents the distance of x from zero on the number line, always yielding a non-negative value.

Definition:

|x| = x if x ≥ 0
|x| = -x if x < 0

Key Properties:

|x| ≥ 0 for all x.
|-x| = |x|
|xy| = |x||y|
|x/y| = |x|/|y| (for y ≠ 0)
Triangle Inequality: |a + b| ≤ |a| + |b|

Graphical Representation: The graph of y = |x| forms a 'V' shape, symmetric about the y-axis, with its vertex at the origin. For y = |f(x)|, any part of the graph of y = f(x) that is below the x-axis is reflected above the x-axis.

Solving Equations with Modulus Functions

Equations involving the modulus function can be solved by considering two cases or by squaring both sides.

Method 1: ...

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Solving Inequalities with Modulus Functions

Modulus inequalities require careful consideration of cases or graphical interpretation.

**Type 1: |f(x)| < k (or ≤ ...**

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Exam Tips

1.For quadratic inequalities, always sketch the parabola or use a sign diagram to correctly identify the solution intervals. Don't just rely on algebraic manipulation.
2.When solving modulus equations/inequalities by squaring both sides, be careful not to introduce extraneous solutions. Always check your final answers in the original equation/inequality, especially if one side was not guaranteed to be positive.
3.Remember the definition of the modulus function: `|x| = x` for `x ≥ 0` and `|x| = -x` for `x < 0`. This is crucial for solving problems by considering cases.
4.Practice solving inequalities with fractions (rational inequalities). Remember to find critical values from both the numerator and denominator, and be mindful of values that make the denominator zero.
5.Clearly show your working steps, especially when dealing with multiple cases or complex algebraic manipulations. Partial credit is awarded for correct methods.

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