Identities and Inequalities (OLevel's Pure Math)

1. Algebraic Identities

An identity is a statement that is true for all values of the variable. Unlike a standard equation, we use the identity symbol $\equiv$.

Equating Coefficients

If two polynomials are identical, the coefficients of corresponding powers of $x$ must be equal. This is the most common method for finding unknown constants.

Example: Find $A$ and $B$ if $3x+7\equiv A(x-1)+B(x+2)$.

• Method 1: Substitution. Let $x=1$ to eliminate $A$. Let $x=-2$ to eliminate $B$.
• Method 2: Equating. Expand the right side: $(A+B)x+(-A+2B)$. Then: $A+B=3$ and $-A+2B=7$.

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2. Linear Inequalities

Solving linear inequalities is similar to solving equations, with one golden rule:

When you multiply or divide by a negative number, you must reverse the inequality sign.

$$-2x<10\Rightarrow x>-5$$

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3. Quadratic Inequalities

Quadratic inequalities cannot be solved by simple rearrangement. You must follow these steps:

1. Rearrange: Make one side of the inequality zero.
2. Find Critical Values: Solve the quadratic as an equation (factorize or use the formula).
3. Sketch the Curve: Draw a quick parabola based on the roots and the sign of $x^2$.
4. Identify the Region: Look at the graph to see where it is above or below the x-axis.

Inequality Type	Region Required	Resulting Form (if roots $\alpha <\beta$)
$f\left(x\right)>0$	Above the x-axis	$x<\alpha$ or $x>\beta$
$f\left(x\right)<0$	Below the x-axis	$\alpha <x<\beta$

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4. Worked Example: Quadratic Inequality

Question: Solve $x^2-2x-8\ge 0$.

Step 1: Find Critical Values
Factorize: $(x-4)(x+2)=0$.
Critical values are $x=4$ and $x=-2$.

Step 2: Sketch
Since the $x^2$ coefficient is positive, it's a "happy" parabola crossing at -2 and 4.

Step 3: Identify the Solution
We want $\ge 0$ (the parts above or on the x-axis).
Final Answer: $x\le -2$ or $x\ge 4$.

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5. Set Notation in Inequalities

Examiners often ask for the solution set. For the example above, you would write:

$$\{x:x\le -2\}\cup \{x:x\ge 4\}$$