Inequalities

Hey students! 👋 Welcome to our exploration of inequalities - one of the most powerful tools in mathematics that helps us describe ranges of values and solve real-world problems. In this lesson, you'll master solving linear, polynomial, rational, and absolute value inequalities while learning to represent solutions using number lines and interval notation. By the end, you'll be able to tackle everything from determining safe driving speeds to calculating profit margins for businesses! 🚗💰

Understanding Inequalities and Their Real-World Applications

An inequality is a mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥. Unlike equations that have specific solutions, inequalities describe ranges of values that make the statement true. Think of inequalities as describing "zones" rather than exact points!

Let's start with a real-world example: Imagine you're planning a road trip and your car gets 25 miles per gallon. If you have $40 to spend on gas at $4 per gallon, how many miles can you travel? You can buy 10 gallons of gas, so you can travel at most 250 miles. This creates the inequality: distance ≤ 250 miles.

The four main inequality symbols are:

< means "less than"
> means "greater than"
≤ means "less than or equal to"
≥ means "greater than or equal to"

When we solve inequalities, we're finding all values that make the inequality statement true. The solution is typically a range of numbers, which we can represent on a number line or using interval notation.

Linear Inequalities: The Foundation

Linear inequalities are the simplest type and follow patterns similar to linear equations. The key difference is that when you multiply or divide both sides by a negative number, you must flip the inequality sign! This is crucial and a common source of mistakes.

Let's solve the inequality $3x - 7 > 2x + 5$:

First, subtract $2x$ from both sides: $x - 7 > 5$

Then add 7 to both sides: $x > 12$

The solution includes all numbers greater than 12. On a number line, we'd draw an open circle at 12 (since 12 isn't included) and shade everything to the right. In interval notation, this is written as $(12, ∞)$.

Here's where the sign-flipping rule comes into play. Consider $-2x + 6 ≥ 10$:

Subtract 6: $-2x ≥ 4$

Divide by -2 (flip the sign!): $x ≤ -2$

Real-world application: A delivery company charges $15 plus $3 per mile. If you have at most $45 to spend, how far can they deliver? The inequality is $15 + 3x ≤ 45$, which gives us $x ≤ 10$ miles.

Polynomial Inequalities: Working with Curves

Polynomial inequalities involve expressions with variables raised to powers greater than 1. These create curved graphs, making the solution process more complex. The key strategy is finding where the polynomial equals zero (critical points) and testing intervals between these points.

Consider the inequality $x^2 - 5x + 6 > 0$. First, we factor: $(x - 2)(x - 3) > 0$.

The critical points are $x = 2$ and $x = 3$ (where the expression equals zero). These points divide the number line into three intervals: $(-∞, 2)$, $(2, 3)$, and $(3, ∞)$.

Now we test a point in each interval:

For $x = 0$ (in the first interval): $(0-2)(0-3) = 6 > 0$ ✓
For $x = 2.5$ (in the middle): $(2.5-2)(2.5-3) = -0.25 < 0$ ✗
For $x = 4$ (in the last interval): $(4-2)(4-3) = 2 > 0$ ✓

Therefore, the solution is $x < 2$ or $x > 3$, written as $(-∞, 2) ∪ (3, ∞)$.

This type of inequality appears in physics when analyzing projectile motion. If a ball's height is given by $h = -16t^2 + 64t$, when is the ball above 48 feet? We solve $-16t^2 + 64t > 48$, which gives us the time interval when the ball is at that height.

Rational Inequalities: Dealing with Fractions

Rational inequalities contain fractions with variables in the denominator. The critical difference from polynomial inequalities is that we must consider where the denominator equals zero (undefined points) as well as where the numerator equals zero.

Let's solve $\frac{x + 1}{x - 2} ≥ 0$.

Critical points occur when:

Numerator = 0: $x + 1 = 0$, so $x = -1$
Denominator = 0: $x - 2 = 0$, so $x = 2$

These points create intervals: $(-∞, -1)$, $(-1, 2)$, and $(2, ∞)$.

Testing each interval:

$x = -2$: $\frac{-1}{-4} = \frac{1}{4} > 0$ ✓
$x = 0$: $\frac{1}{-2} = -\frac{1}{2} < 0$ ✗
$x = 3$: $\frac{4}{1} = 4 > 0$ ✓

Since we want ≥ 0, we include $x = -1$ (where the expression equals 0) but exclude $x = 2$ (where it's undefined).

Solution: $x ∈ [-1, 2) ∪ (2, ∞)$ or $x ≤ -1$ or $x > 2$.

A practical example: If the average cost per item is $\frac{1000 + 5x}{x}$ dollars, when is this cost at most $15? We solve $\frac{1000 + 5x}{x} ≤ 15$, helping businesses determine optimal production levels.

Absolute Value Inequalities: Distance and Tolerance

Absolute value inequalities describe distances from zero or tolerance ranges. There are two main types with different solution patterns.

Type 1: $|x| < a$ (where $a > 0$)

This means the distance from zero is less than $a$, so: $-a < x < a$

Example: $|x - 3| < 5$ means the distance from 3 is less than 5.

This gives us: $-5 < x - 3 < 5$

Adding 3: $-2 < x < 8$ or $x ∈ (-2, 8)$

Type 2: $|x| > a$ (where $a > 0$)

This means the distance from zero is greater than $a$, so: $x < -a$ or $x > a$

Example: $|2x + 1| ≥ 7$ gives us two cases:

$2x + 1 ≥ 7$, so $x ≥ 3$
$2x + 1 ≤ -7$, so $x ≤ -4$

Solution: $x ∈ (-∞, -4] ∪ [3, ∞)$

Manufacturing applications: If a machine produces bolts that must be 2.5 ± 0.1 inches long, the acceptable range is $|length - 2.5| ≤ 0.1$, giving us bolts between 2.4 and 2.6 inches long.

Conclusion

Inequalities are powerful mathematical tools that help us describe ranges, set boundaries, and solve real-world optimization problems. You've learned to solve linear inequalities (remembering to flip signs when multiplying by negatives), polynomial inequalities (using critical points and interval testing), rational inequalities (watching for undefined points), and absolute value inequalities (understanding distance interpretations). These skills will serve you well in advanced mathematics, science courses, and practical problem-solving situations throughout your academic and professional career! 🎯

Study Notes

• Linear Inequalities: Solve like equations, but flip the inequality sign when multiplying or dividing by negative numbers

• Polynomial Inequalities: Find critical points where expression equals zero, test intervals between points

• Rational Inequalities: Consider both numerator = 0 and denominator = 0 as critical points; exclude undefined points from solutions

• Absolute Value Type 1: $|x| < a$ becomes $-a < x < a$ (intersection/and)

• Absolute Value Type 2: $|x| > a$ becomes $x < -a$ or $x > a$ (union/or)

• Number Line Notation: Open circles for < or >, closed circles for ≤ or ≥

• Interval Notation: Use parentheses ( ) for open intervals, brackets [ ] for closed intervals

• Critical Points: Values where expression equals zero or becomes undefined

• Interval Testing: Choose test points between critical points to determine sign of expression

• Union Symbol: ∪ means "or" - combines separate solution intervals