Inequalities: A Quick Guide

Understanding Inequalities 🤔

Inequalities are mathematical expressions that show the relationship between two values that are not equal. Unlike equations, which use an equals sign (=), inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Basic Inequality Symbols ➕

• <: Less than (e.g., $x < 5$ means x is less than 5)
• >: Greater than (e.g., $y > 3$ means y is greater than 3)
• ≤: Less than or equal to (e.g., $a ≤ 7$ means a is less than or equal to 7)
• ≥: Greater than or equal to (e.g., $b ≥ 10$ means b is greater than or equal to 10)
• ≠: Not equal to (e.g., $c ≠ 2$ means c is not equal to 2)

Solving Linear Inequalities ⚙️

Solving inequalities is similar to solving equations, but there are a few key differences. One important rule is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Example:

Solve the inequality: $3x - 2 > 7$

1. Add 2 to both sides: $3x > 9$
2. Divide both sides by 3: $x > 3$

So, the solution is $x > 3$, which means x can be any number greater than 3.

Multiplying/Dividing by a Negative Number:

Solve the inequality: $-2x ≤ 8$

1. Divide both sides by -2 (and reverse the inequality sign): $x ≥ -4$

The solution is $x ≥ -4$.

Graphing Inequalities 📈

Inequalities can be represented graphically on a number line. Here’s how:

• For $x > a$ or $x < a$, use an open circle at 'a' to indicate that 'a' is not included in the solution.
• For $x ≥ a$ or $x ≤ a$, use a closed circle at 'a' to indicate that 'a' is included in the solution.
• Shade the number line in the direction of the solution.

Example:

Graph $x ≤ 2$

Place a closed circle at 2 on the number line and shade to the left, indicating all numbers less than or equal to 2.

Compound Inequalities 🔗

Compound inequalities involve two or more inequalities combined.

"And" Inequalities:

These are inequalities where both conditions must be true. For example, $2 < x < 5$ means x is greater than 2 and less than 5.

"Or" Inequalities:

These are inequalities where at least one condition must be true. For example, $x < 1$ or $x > 4$ means x is less than 1 or greater than 4.

Absolute Value Inequalities 🧮

Absolute value inequalities involve absolute value expressions. Remember that absolute value represents the distance from zero.

Example:

Solve $|x| < 3$

This means x is within 3 units of zero, so $-3 < x < 3$.

Real-World Applications 🌍

Inequalities are used in various real-world scenarios:

• Budgeting: Determining how much you can spend without exceeding your budget.
• Engineering: Ensuring structures can withstand certain loads.
• Optimization: Finding the best possible solution within certain constraints.