Linear Inequalities and Absolute Value
Summary
This chapter extends equation-solving techniques to inequalities and absolute value situations. Students will learn to solve and graph linear inequalities, understanding how solution sets differ from equations and how to represent solutions on a number line. The chapter covers compound inequalities using "and" and "or" logic, and introduces absolute value equations and inequalities, helping students understand the geometric meaning of absolute value and solve more complex algebraic problems.
Concepts Covered
This chapter covers the following 10 concepts from the learning graph:
- Inequality Symbols
- Linear Inequality
- Solving Linear Inequalities
- Compound Inequalities
- And Inequalities
- Or Inequalities
- Absolute Value Equations
- Absolute Value Inequalities
- Graphing Inequalities
- Applications of Linear Equations
Prerequisites
This chapter builds on concepts from:
- Chapter 1: Foundations of Algebra
- Chapter 2: Number Systems and Properties
- Chapter 6: Solving Linear Equations
Understanding Inequalities
An inequality is a mathematical statement that compares two expressions using inequality symbols rather than an equals sign. While equations state that two expressions are equal, inequalities describe relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another.
Inequality Symbols
There are four main inequality symbols you need to know:
| Symbol | Meaning | Example | Read as |
|---|---|---|---|
| \(<\) | Less than | \(x < 5\) | "\(x\) is less than 5" |
| \(>\) | Greater than | \(x > -2\) | "\(x\) is greater than \(-2\)" |
| \(\leq\) | Less than or equal to | \(x \leq 10\) | "\(x\) is less than or equal to 10" |
| \(\geq\) | Greater than or equal to | \(x \geq 0\) | "\(x\) is greater than or equal to 0" |
Key difference from equations: - An equation like \(x = 5\) has one solution: 5 - An inequality like \(x < 5\) has infinitely many solutions: 4, 3.5, 0, -10, etc.
Graphing Inequalities on a Number Line
Graphing inequalities helps visualize all the solutions. We use a number line to show which values satisfy the inequality.
Graphing rules: - Open circle (○) for \(<\) or \(>\) (the endpoint is NOT included) - Closed circle (●) for \(\leq\) or \(\geq\) (the endpoint IS included) - Shade in the direction of the solutions
Examples:
\(x > 3\): Open circle at 3, shade to the right
1 2 | |
\(x \leq -1\): Closed circle at -1, shade to the left
1 2 | |
\(x \geq 2\): Closed circle at 2, shade to the right
1 2 | |
\(x < 0\): Open circle at 0, shade to the left
1 2 | |
Solving Linear Inequalities
A linear inequality is similar to a linear equation, but with an inequality symbol instead of an equals sign.
The process for solving linear inequalities is almost identical to solving linear equations, with one crucial exception.
Basic Solving Rules
You can perform the same operations on both sides of an inequality as you do with equations:
✓ Add or subtract the same number from both sides ✓ Multiply or divide both sides by a positive number
EXCEPTION: When you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
Why reverse the symbol?
Consider: \(3 > 1\) (true)
Multiply both sides by −1:
\(-3 \stackrel{?}{>} -1\) (false!)
We need to reverse the symbol: \(-3 < -1\) (true!)
Solving One-Step and Two-Step Inequalities
Example 1: Solve \(x + 5 > 12\)
Subtract 5 from both sides:
\(x > 7\)
Graph: Open circle at 7, shade right
Solution set in interval notation: \((7, \infty)\)
Example 2: Solve \(3x \leq 15\)
Divide both sides by 3:
\(x \leq 5\)
Graph: Closed circle at 5, shade left
Solution set: \((-\infty, 5]\)
Example 3: Solve \(-2x > 10\)
Divide both sides by −2 and reverse the symbol:
\(x < -5\)
Graph: Open circle at −5, shade left
Solution set: \((-\infty, -5)\)
Example 4: Solve \(\frac{x}{4} - 3 \geq 2\)
Step 1: Add 3 to both sides
\(\frac{x}{4} \geq 5\)
Step 2: Multiply both sides by 4
\(x \geq 20\)
Solution set: \([20, \infty)\)
Solving Multi-Step Inequalities
The process is the same as for equations: simplify, collect like terms, and isolate the variable.
Example 1: Solve \(3x - 7 < 5x + 9\)
Subtract \(3x\) from both sides:
\(-7 < 2x + 9\)
Subtract 9 from both sides:
\(-16 < 2x\)
Divide by 2:
\(-8 < x\)
Or equivalently: \(x > -8\)
Solution set: \((-8, \infty)\)
Example 2: Solve \(-3(x + 2) \geq 12\)
Distribute:
\(-3x - 6 \geq 12\)
Add 6:
\(-3x \geq 18\)
Divide by −3 and reverse:
\(x \leq -6\)
Solution set: \((-\infty, -6]\)
Example 3: Solve \(\frac{2x - 5}{3} < 7\)
Multiply both sides by 3:
\(2x - 5 < 21\)
Add 5:
\(2x < 26\)
Divide by 2:
\(x < 13\)
Solution set: \((-\infty, 13)\)
Compound Inequalities
A compound inequality consists of two inequalities joined by "and" or "or."
And Inequalities
And inequalities require both conditions to be true simultaneously. The solution is the intersection of the two solution sets.
Form: \(a < x < b\) (meaning \(x > a\) AND \(x < b\))
Example 1: Solve \(-3 < x + 2 < 7\)
This means: \(x + 2 > -3\) AND \(x + 2 < 7\)
Solve both parts by subtracting 2:
\(-5 < x\) AND \(x < 5\)
Combined: \(-5 < x < 5\)
Graph: Open circles at −5 and 5, shade between them
1 2 | |
Solution set: \((-5, 5)\)
Example 2: Solve \(2 \leq 3x - 1 \leq 11\)
Add 1 to all parts:
\(3 \leq 3x \leq 12\)
Divide all parts by 3:
\(1 \leq x \leq 4\)
Solution set: \([1, 4]\)
Graph: Closed circles at 1 and 4, shade between
Or Inequalities
Or inequalities require at least one condition to be true. The solution is the union of the two solution sets.
Example 1: Solve \(x < -2\) OR \(x > 5\)
These are already solved. Graph both:
Graph: Open circle at −2, shade left; open circle at 5, shade right
1 2 | |
Solution set: \((-\infty, -2) \cup (5, \infty)\)
Example 2: Solve \(2x + 1 < -5\) OR \(3x - 4 > 8\)
Solve the first inequality:
\(2x < -6\) \(x < -3\)
Solve the second inequality:
\(3x > 12\) \(x > 4\)
Solution set: \((-\infty, -3) \cup (4, \infty)\)
Example 3: Solve \(x + 3 \leq 1\) OR \(x - 2 \geq 5\)
First: \(x \leq -2\) Second: \(x \geq 7\)
Solution set: \((-\infty, -2] \cup [7, \infty)\)
Absolute Value Equations
The absolute value of a number is its distance from zero on the number line, always non-negative.
Notation: \(|x|\) read as "the absolute value of \(x\)"
Examples: - \(|5| = 5\) - \(|-5| = 5\) - \(|0| = 0\)
Solving Absolute Value Equations
Key principle: If \(|x| = a\) where \(a > 0\), then \(x = a\) or \(x = -a\).
This is because both \(a\) and \(-a\) are distance \(a\) from zero.
Example 1: Solve \(|x| = 7\)
\(x = 7\) or \(x = -7\)
Check: \(|7| = 7\) ✓ and \(|-7| = 7\) ✓
Example 2: Solve \(|x + 3| = 5\)
Set up two equations:
\(x + 3 = 5\) or \(x + 3 = -5\)
Solve each:
\(x = 2\) or \(x = -8\)
Check: \(|2 + 3| = |5| = 5\) ✓ and \(|-8 + 3| = |-5| = 5\) ✓
Example 3: Solve \(|2x - 1| = 9\)
\(2x - 1 = 9\) or \(2x - 1 = -9\)
First equation: \(2x = 10\), so \(x = 5\)
Second equation: \(2x = -8\), so \(x = -4\)
Solutions: \(x = 5\) or \(x = -4\)
Example 4: Solve \(3|x - 2| + 5 = 20\)
First, isolate the absolute value:
\(3|x - 2| = 15\)
\(|x - 2| = 5\)
Now solve:
\(x - 2 = 5\) or \(x - 2 = -5\)
\(x = 7\) or \(x = -3\)
Special Cases
If \(|x| = 0\): The only solution is \(x = 0\)
If \(|x| = -5\): No solution (absolute value cannot be negative)
Example: Solve \(|x + 4| = -2\)
No solution (the empty set: \(\emptyset\))
Absolute Value Inequalities
Absolute value inequalities involve absolute value expressions with inequality symbols.
Less Than Type: \(|x| < a\)
If \(|x| < a\) (where \(a > 0\)), this means \(x\) is less than \(a\) units from zero.
Solution: \(-a < x < a\)
Think of it as: \(x\) is between \(-a\) and \(a\)
Example 1: Solve \(|x| < 4\)
\(-4 < x < 4\)
Graph: Open circles at −4 and 4, shade between
Solution set: \((-4, 4)\)
Example 2: Solve \(|x + 1| \leq 5\)
\(-5 \leq x + 1 \leq 5\)
Subtract 1 from all parts:
\(-6 \leq x \leq 4\)
Solution set: \([-6, 4]\)
Example 3: Solve \(|3x - 2| < 7\)
\(-7 < 3x - 2 < 7\)
Add 2 to all parts:
\(-5 < 3x < 9\)
Divide by 3:
\(-\frac{5}{3} < x < 3\)
Solution set: \(\left(-\frac{5}{3}, 3\right)\)
Greater Than Type: \(|x| > a\)
If \(|x| > a\) (where \(a > 0\)), this means \(x\) is more than \(a\) units from zero.
Solution: \(x < -a\) OR \(x > a\)
Example 1: Solve \(|x| > 3\)
\(x < -3\) OR \(x > 3\)
Graph: Open circle at −3 shade left, open circle at 3 shade right
Solution set: \((-\infty, -3) \cup (3, \infty)\)
Example 2: Solve \(|x - 4| \geq 2\)
\(x - 4 \leq -2\) OR \(x - 4 \geq 2\)
\(x \leq 2\) OR \(x \geq 6\)
Solution set: \((-\infty, 2] \cup [6, \infty)\)
Example 3: Solve \(|2x + 5| > 9\)
\(2x + 5 < -9\) OR \(2x + 5 > 9\)
First: \(2x < -14\), so \(x < -7\)
Second: \(2x > 4\), so \(x > 2\)
Solution set: \((-\infty, -7) \cup (2, \infty)\)
Summary of Absolute Value Inequalities
| Type | Compound Form | Graph Type |
|---|---|---|
| \(\|x\| < a\) | \(-a < x < a\) | AND (between two values) |
| \(\|x\| \leq a\) | \(-a \leq x \leq a\) | AND (between, including endpoints) |
| \(\|x\| > a\) | \(x < -a\) OR \(x > a\) | OR (outside two values) |
| \(\|x\| \geq a\) | \(x \leq -a\) OR \(x \geq a\) | OR (outside, including endpoints) |
Applications of Linear Equations and Inequalities
Inequalities model many real-world situations where there are ranges of acceptable values rather than exact values.
Example 1: Temperature Range
The temperature in a refrigerator must stay between 35°F and 40°F. Write and solve an inequality.
Let \(T\) = temperature
\(35 \leq T \leq 40\)
This is already solved. The temperature must be in the range \([35, 40]\).
Example 2: Budget Constraint
You have $50 to spend on books. Each book costs $12. How many books can you buy?
Let \(n\) = number of books
\(12n \leq 50\)
\(n \leq \frac{50}{12} = 4.17...\)
Since \(n\) must be a whole number, you can buy at most 4 books.
Example 3: Profit Inequality
A company's profit is given by \(P = 15x - 200\), where \(x\) is the number of items sold. How many items must be sold to make a profit of at least $1000?
\(15x - 200 \geq 1000\)
\(15x \geq 1200\)
\(x \geq 80\)
At least 80 items must be sold.
Example 4: Distance Tolerance
A machine part must be within 0.02 mm of 5 mm to be acceptable. Write this as an absolute value inequality.
Let \(d\) = actual dimension
The distance from 5 must be at most 0.02:
\(|d - 5| \leq 0.02\)
Solve:
\(-0.02 \leq d - 5 \leq 0.02\)
\(4.98 \leq d \leq 5.02\)
The part must be between 4.98 mm and 5.02 mm.
Summary
In this chapter, you've learned to work with inequalities and absolute values:
Inequalities:
- Symbols: \(<\), \(>\), \(\leq\), \(\geq\)
- Graphing: Use open circles for \(<\) and \(>\), closed circles for \(\leq\) and \(\geq\)
- Solving: Same as equations, but reverse the inequality when multiplying/dividing by a negative number
- Compound inequalities:
- AND: Solutions satisfy both conditions (intersection)
- OR: Solutions satisfy at least one condition (union)
Absolute Value:
- Equations: \(|x| = a\) gives \(x = a\) or \(x = -a\)
- Less than inequalities: \(|x| < a\) gives \(-a < x < a\) (AND)
- Greater than inequalities: \(|x| > a\) gives \(x < -a\) OR \(x > a\)
Key Skills:
- Solve linear inequalities and graph solutions
- Solve compound inequalities with "and" or "or"
- Solve absolute value equations and inequalities
- Apply inequalities to real-world constraints
These tools allow you to model and solve problems involving ranges, tolerances, and conditions—essential skills for mathematics, science, engineering, and everyday decision-making.
References
- Inequalities on a Number Line - Steps, Examples & Questions - 2024 - Third Space Learning - Visual guide to graphing linear inequalities on number lines with clear explanations of open vs. closed circles and direction of shading, addressing common student challenges.