## AI Inequalities

MathCrave AI inequalities solver examines and resolves various inequality equations using advanced AI algorithms and techniques. This solver has the capability to tackle a diverse range of inequality problems, such as linear inequalities, quadratic inequalities, absolute value inequalities, and even more intricate types.

### Lesson Notes: Inequalities

#### 1. Simple Rules for Inequalities

Basic Inequality Symbols:

– \( > \): Greater than

– \( < \): Less than

– \( \geq \): Greater than or equal to

– \( \leq \): Less than or equal to

**Rules**:

**1. Addition/Subtraction Rule:**

– If \( a > b \), then \( a + c > b + c \).

– If \( a < b \), then \( a – c < b – c \).

**2. Multiplication/Division Rule:**

– If \( a > b \) and \( c > 0 \), then \( ac > bc \).

– If \( a > b \) and \( c < 0 \), then \( ac < bc \).

– If \( a > b \) and \( c > 0 \), then \( \frac{a}{c} > \frac{b}{c} \).

– If \( a > b \) and \( c < 0 \), then \( \frac{a}{c} < \frac{b}{c} \).

#### 2. Simple Inequalities

Example: Solve \( 3x + 5 > 11 \).

Steps:

1. Subtract 5 from both sides: \( 3x > 6 \).

2. Divide by 3: \( x > 2 \).

Solution: \( x > 2 \).

#### 3. Inequalities Involving a Modulus

Example: Solve \( |x – 3| < 5 \).

Steps:

1. Break into two inequalities: \( -5 < x – 3 < 5 \).

2. Add 3 to all parts: \( -2 < x < 8 \).

Solution: \( -2 < x < 8 \).

#### 4. Inequalities Involving Quotients

Example: Solve \( \frac{x + 1}{x – 2} > 1 \).

Steps:

1. Move 1 to the other side: \( \frac{x + 1}{x – 2} – 1 > 0 \).

2. Combine into a single fraction: \( \frac{x + 1 – (x – 2)}{x – 2} > 0 \).

3. Simplify: \( \frac{3}{x – 2} > 0 \).

Solution:

\[ x – 2 > 0 \implies x > 2 \]

Thus, \( x > 2 \).

#### 5. Inequalities Involving Square Functions

Example: Solve \( (x – 1)^2 \leq 4 \).

Steps:

1. Take the square root of both sides: \( -2 \leq x – 1 \leq 2 \).

2. Add 1 to all parts: \( -1 \leq x \leq 3 \).

Solution: \( -1 \leq x \leq 3 \).

#### 6. Quadratic Inequalities

Example: Solve \( x^2 – 5x + 6 < 0 \).

Steps:

1. Factor the quadratic: \( (x – 2)(x – 3) < 0 \).

2. Determine the critical points: \( x = 2 \) and \( x = 3 \).

3. Test intervals around the critical points:

– For \( x < 2 \), choose \( x = 1 \): \( (1 – 2)(1 – 3) = (-1)(-2) > 0 \) (false).

– For \( 2 < x < 3 \), choose \( x = 2.5 \): \( (2.5 – 2)(2.5 – 3) = (0.5)(-0.5) < 0 \) (true).

– For \( x > 3 \), choose \( x = 4 \): \( (4 – 2)(4 – 3) = (2)(1) > 0 \) (false).

Solution: \( 2 < x < 3 \).