What is the definition of an inequality?

An inequality is a mathematical statement that shows the relationship between two expressions that are not necessarily equal. 
It uses symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

If a>b and b>c, what can you conclude about a and c?

a > c (Inequalities are transitive).

What happens when you multiply or divide both sides of an inequality by a negative number?

The inequality sign reverses direction. For example, if a>b, then −a < −b.

State the AM-GM inequality for two positive numbers a and b.

with equality if and only if a=b.

How do you solve the inequality 

Find the critical points x = 3 and x = -2. Test intervals:

• For x < -2, denominator negative, numerator negative - positive? No.
• For -2 < x < 3, denominator positive, numerator negative - negative.
• For x > 3, denominator positive, numerator positive - positive.

Given x > 0, prove 

By AM-GM: 

If a, b, c > 0, prove

By AM-GM on the terms

Solve the inequality:
∣2x − 5∣ ≤ 7.

7 ≤ 2x - 5 ≤ 7
Add 5: 2 ≤ 2x ≤ 12
Divide by 2: -1 ≤ x ≤ 6.

What is the Cauchy-Schwarz inequality in vector form?

For vectors u, v, |u . v| ≤ || u|| ||v||.

State the rearrangement inequality conceptually.

For two sequences sorted in ascending order, the sum of the products of corresponding terms is maximized; sum of products of opposite order is minimized.

Prove that for all real x, y, (x - y)² ≥ 0 leads to an inequality.

Expand: x² - 2xy + y² ≥ 0 ⇒ x² + y² ≥ 2xy.

 Given a + b = 1, with a, b > 0, find the minimum value of 

Use AM-HM or rewrite: Maximize ab by AM-GM:
So minimum

Solve for 

Rearrange:

Critical points: x=−4, x = 1.
Test intervals: x < -4: numerator negative, denominator negative - positive
-4 < x < 1: numerator positive, denominator negative - negative
x > 1: numerator positive, denominator positive - positive

What is the key property of quadratic inequalities?

For ax² + bx + c > 0, the solution depends on the sign of a and the roots of the quadratic.

Prove that if a, b, c > 0, then a³ + b³ + c³ ≥ 3abc.

By AM-GM on cubes: 

If x > 0, compare x + 1/x and 2.

by AM-GM equality.

Solve: ∣x−3∣>5.

 x - 3 > 5 ⇒ 8 or x - 3 <- 5 ⇒ x < -2.

What is the solution set for  x² − 4x + 3 < 0?

Factor: (x−1)(x−3)<0, solution is 1< x < 3.

State the inequality involving sums and reciprocals: For positive a, b, prove 

This is the AM-GM inequality.

Given a, b > 0, prove 

By AM-GM:  so sum ≥ 2.