MATHCRAVE
MathCrave Math
Hi! How can MathCrave help you today on Inequalities?

# AI Inequalities

## 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:

– 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: $$2 < x < 3$$.