Chapter Notes: Expressions, Equations, & Inequalities
Algebra is the language of mathematics that lets us describe patterns, solve problems, and find unknown values. In this chapter, you will learn to work with expressions, which are mathematical phrases; equations, which are mathematical sentences that state two things are equal; and inequalities, which compare quantities that are not necessarily equal. These tools help us solve real-world problems like calculating costs, finding distances, and determining how much of something we need.
Understanding Expressions
An expression is a mathematical phrase made up of numbers, variables, and operations. Unlike an equation, an expression does not have an equals sign. You can think of an expression as a recipe that tells you what calculations to perform, but it doesn't tell you what the final result equals.
Examples of expressions include \( 3x + 5 \), \( 2y - 7 \), and \( 4a^2 + 3a - 2 \). In these expressions, the letters (like \( x \), \( y \), and \( a \)) are called variables because their values can vary or change. The numbers next to the variables, like the 3 in \( 3x \), are called coefficients. Numbers that stand alone, like the 5 in \( 3x + 5 \), are called constants.
Parts of an Expression
To work effectively with expressions, you need to understand their components:
- Variables: Letters that represent unknown or changing quantities
- Coefficients: Numbers multiplied by variables
- Constants: Numbers that stand alone and don't change
- Terms: Parts of an expression separated by addition or subtraction signs
- Operations: Addition (+), subtraction (-), multiplication (×), and division (÷)
In the expression \( 5x + 3y - 8 \), there are three terms: \( 5x \), \( 3y \), and \( -8 \). The first term has a coefficient of 5 and a variable \( x \). The second term has a coefficient of 3 and a variable \( y \). The third term is a constant, -8.
Evaluating Expressions
To evaluate an expression means to find its numerical value when you know the value of each variable. You substitute the given numbers for the variables and then perform the operations following the order of operations.
Example: Evaluate the expression \( 4x + 7 \) when \( x = 3 \).
What is the value of the expression?
Solution:
Substitute 3 for \( x \) in the expression:
\( 4(3) + 7 \)
Multiply first: \( 12 + 7 \)
Add: \( 12 + 7 = 19 \)
The value of the expression is 19.
Example: Evaluate \( 3a^2 - 2a + 5 \) when \( a = 4 \).
What is the value of the expression?
Solution:
Substitute 4 for \( a \):
\( 3(4)^2 - 2(4) + 5 \)
Calculate the exponent: \( 3(16) - 2(4) + 5 \)
Multiply: \( 48 - 8 + 5 \)
Subtract and add from left to right: \( 40 + 5 = 45 \)
The value of the expression is 45.
Simplifying Expressions
To simplify an expression means to write it in the shortest or most compact form possible. You do this by combining like terms. Like terms are terms that have exactly the same variable raised to the same power. For example, \( 3x \) and \( 5x \) are like terms, but \( 3x \) and \( 3y \) are not.
When combining like terms, you add or subtract only the coefficients while keeping the variable part the same.
Example: Simplify the expression \( 7x + 3x - 2 \).
What is the simplified form?
Solution:
Identify like terms: \( 7x \) and \( 3x \) are like terms
Combine the coefficients: \( 7 + 3 = 10 \)
Keep the variable: \( 10x - 2 \)
The simplified expression is \( 10x - 2 \).
Example: Simplify \( 5a + 2b - 3a + 4b + 1 \).
What is the simplified form?
Solution:
Group like terms together: \( (5a - 3a) + (2b + 4b) + 1 \)
Combine the \( a \) terms: \( 5a - 3a = 2a \)
Combine the \( b \) terms: \( 2b + 4b = 6b \)
Write the simplified expression: \( 2a + 6b + 1 \)
The simplified expression is \( 2a + 6b + 1 \).
Working with Equations
An equation is a mathematical sentence that states two expressions are equal. Every equation has an equals sign (=) that separates the left side from the right side. For example, \( x + 5 = 12 \) is an equation. The expression on the left is \( x + 5 \), and the expression on the right is \( 12 \).
The goal when solving an equation is to find the value of the variable that makes the equation true. This value is called the solution of the equation.
One-Step Equations
A one-step equation can be solved with just one operation. To solve these equations, you perform the inverse (opposite) operation on both sides of the equation.
- If the equation involves addition, subtract the same number from both sides
- If the equation involves subtraction, add the same number to both sides
- If the equation involves multiplication, divide both sides by the same number
- If the equation involves division, multiply both sides by the same number
Example: Solve the equation \( x + 9 = 15 \).
What value of \( x \) makes this equation true?
Solution:
The variable \( x \) has 9 added to it, so subtract 9 from both sides:
\( x + 9 - 9 = 15 - 9 \)
Simplify: \( x = 6 \)
The solution is \( x = 6 \).
Example: Solve \( 4y = 28 \).
What value of \( y \) makes this equation true?
Solution:
The variable \( y \) is multiplied by 4, so divide both sides by 4:
\( \frac{4y}{4} = \frac{28}{4} \)
Simplify: \( y = 7 \)
The solution is \( y = 7 \).
Two-Step Equations
A two-step equation requires two operations to solve. The general strategy is to undo addition or subtraction first, then undo multiplication or division. This follows the reverse of the order of operations.
Example: Solve the equation \( 3x + 7 = 22 \).
What is the value of \( x \)?
Solution:
First, subtract 7 from both sides to isolate the term with the variable:
\( 3x + 7 - 7 = 22 - 7 \)
\( 3x = 15 \)
Next, divide both sides by 3:
\( \frac{3x}{3} = \frac{15}{3} \)
\( x = 5 \)
The solution is \( x = 5 \).
Example: Solve \( \frac{n}{5} - 3 = 2 \).
What is the value of \( n \)?
Solution:
First, add 3 to both sides:
\( \frac{n}{5} - 3 + 3 = 2 + 3 \)
\( \frac{n}{5} = 5 \)
Next, multiply both sides by 5:
\( 5 \cdot \frac{n}{5} = 5 \cdot 5 \)
\( n = 25 \)
The solution is \( n = 25 \).
Equations with Variables on Both Sides
Some equations have variables on both sides of the equals sign. To solve these, you need to get all the variable terms on one side and all the constant terms on the other side.
Example: Solve \( 5x - 4 = 3x + 8 \).
What is the value of \( x \)?
Solution:
Subtract \( 3x \) from both sides to get all \( x \) terms on the left:
\( 5x - 3x - 4 = 3x - 3x + 8 \)
\( 2x - 4 = 8 \)
Add 4 to both sides:
\( 2x - 4 + 4 = 8 + 4 \)
\( 2x = 12 \)
Divide both sides by 2:
\( x = 6 \)
The solution is \( x = 6 \).
Solving Inequalities
An inequality is a mathematical statement that compares two expressions using one of these symbols:
- \( < \)="" means="" "less="">
- \( > \) means "greater than"
- \( \leq \) means "less than or equal to"
- \( \geq \) means "greater than or equal to"
- \( \neq \) means "not equal to"
For example, \( x + 3 < 10="" \)="" is="" an="" inequality="" that="" says="" "some="" number="" plus="" 3="" is="" less="" than="" 10."="" unlike="" equations,="" which="" usually="" have="" one="" solution,="" inequalities="" typically="" have="" infinitely="" many="">
Solving One-Step and Two-Step Inequalities
Solving inequalities is very similar to solving equations. You use inverse operations to isolate the variable. However, there is one critical difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.
Example: Solve the inequality \( x + 5 > 12 \).
What values of \( x \) make this inequality true?
Solution:
Subtract 5 from both sides:
\( x + 5 - 5 > 12 - 5 \)
\( x > 7 \)
The solution is \( x > 7 \), meaning any number greater than 7 makes the inequality true.
Example: Solve \( 3y - 7 \leq 11 \).
What values of \( y \) satisfy this inequality?
Solution:
Add 7 to both sides:
\( 3y - 7 + 7 \leq 11 + 7 \)
\( 3y \leq 18 \)
Divide both sides by 3 (since 3 is positive, the inequality symbol stays the same):
\( \frac{3y}{3} \leq \frac{18}{3} \)
\( y \leq 6 \)
The solution is \( y \leq 6 \), meaning any number less than or equal to 6 works.
Example: Solve \( -2x > 10 \).
What values of \( x \) satisfy this inequality?
Solution:
Divide both sides by -2:
Because we are dividing by a negative number, reverse the inequality symbol:
\( \frac{-2x}{-2} < \frac{10}{-2}="">
\( x < -5="">
The solution is \( x < -5="">, meaning any number less than -5 makes the inequality true.
Graphing Solutions on a Number Line
Solutions to inequalities can be shown visually on a number line. This helps you see all the possible values that satisfy the inequality.
- Use an open circle (○) for < or=""> to show that the endpoint is not included
- Use a closed circle (●) for ≤ or ≥ to show that the endpoint is included
- Shade or draw an arrow in the direction of all numbers that are solutions
For \( x > 7 \), you would place an open circle at 7 and shade to the right. For \( y \leq 6 \), you would place a closed circle at 6 and shade to the left. For \( x < -5="" \),="" you="" would="" place="" an="" open="" circle="" at="" -5="" and="" shade="" to="" the="">
Writing Equations and Inequalities from Word Problems
One of the most practical skills in algebra is translating real-world situations into mathematical statements. This process is called modeling. You read the problem carefully, identify what quantity is unknown, choose a variable to represent it, and then write an equation or inequality based on the relationships described.
Key Phrases and Their Mathematical Meanings
Example: A number increased by 8 equals 23.
Write and solve an equation to find the number.
Solution:
Let \( n \) represent the unknown number
"Increased by 8" means add 8: \( n + 8 \)
"Equals 23" gives us: \( n + 8 = 23 \)
Subtract 8 from both sides: \( n = 15 \)
The number is 15.
Example: Maria has $50 and wants to buy books that cost $12 each.
She wants to have at least $10 left over.Write and solve an inequality to find how many books she can buy.
Solution:
Let \( b \) represent the number of books
Each book costs $12, so \( 12b \) is the total cost
She starts with $50 and wants at least $10 left: \( 50 - 12b \geq 10 \)
Subtract 50 from both sides: \( -12b \geq 10 - 50 \)
\( -12b \geq -40 \)
Divide by -12 and reverse the inequality: \( b \leq \frac{40}{12} \)
\( b \leq 3.33... \)
Since she can't buy part of a book, Maria can buy at most 3 books.
Properties of Equality and Inequality
When solving equations and inequalities, we use several important properties that allow us to transform statements while keeping them true.
Properties of Equality
- Addition Property of Equality: If \( a = b \), then \( a + c = b + c \)
- Subtraction Property of Equality: If \( a = b \), then \( a - c = b - c \)
- Multiplication Property of Equality: If \( a = b \), then \( a \times c = b \times c \)
- Division Property of Equality: If \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \)
These properties tell us that whatever we do to one side of an equation, we must do to the other side to keep the equation balanced.
Properties of Inequality
The properties for inequalities are similar, with one crucial exception:
- Addition Property: If \( a < b="" \),="" then="" \(="" a="" +="" c="">< b="" +="" c="">
- Subtraction Property: If \( a < b="" \),="" then="" \(="" a="" -="" c="">< b="" -="" c="">
- Multiplication by Positive: If \( a < b="" \)="" and="" \(="" c=""> 0 \), then \( a \times c < b="" \times="" c="">
- Division by Positive: If \( a < b="" \)="" and="" \(="" c=""> 0 \), then \( \frac{a}{c} < \frac{b}{c}="">
- Multiplication by Negative: If \( a < b="" \)="" and="" \(="" c="">< 0="" \),="" then="" \(="" a="" \times="" c=""> b \times c \) (inequality reverses)
- Division by Negative: If \( a < b="" \)="" and="" \(="" c="">< 0="" \),="" then="" \(="" \frac{a}{c}=""> \frac{b}{c} \) (inequality reverses)
Think of the inequality symbol as an alligator's mouth that always wants to eat the bigger number. When you multiply or divide by a negative number, you're flipping the number line, so the alligator has to turn around to keep eating the bigger number.
Applications and Problem Solving
Expressions, equations, and inequalities are powerful tools for solving real-world problems. The key is to read carefully, define your variable clearly, and translate the situation into mathematical language.
Example: A taxi service charges a $3 base fare plus $2 per mile.
Jake has $25 to spend on a taxi ride.What is the maximum number of miles Jake can travel?
Solution:
Let \( m \) represent the number of miles
The total cost is base fare plus cost per mile: \( 3 + 2m \)
Jake has at most $25 to spend: \( 3 + 2m \leq 25 \)
Subtract 3 from both sides: \( 2m \leq 22 \)
Divide both sides by 2: \( m \leq 11 \)
Jake can travel at most 11 miles.
Example: The perimeter of a rectangle is 36 inches.
The length is 4 inches more than the width.Find the dimensions of the rectangle.
Solution:
Let \( w \) represent the width in inches
Then the length is \( w + 4 \) inches
Perimeter formula: \( P = 2l + 2w \)
Substitute known values: \( 36 = 2(w + 4) + 2w \)
Distribute: \( 36 = 2w + 8 + 2w \)
Combine like terms: \( 36 = 4w + 8 \)
Subtract 8: \( 28 = 4w \)
Divide by 4: \( w = 7 \)
Width is 7 inches, length is \( 7 + 4 = 11 \) inches
The rectangle is 7 inches by 11 inches.
Working with expressions, equations, and inequalities gives you the foundation for all future algebra. These skills allow you to model situations, make predictions, and solve problems efficiently. Whether you're calculating how much money you'll save, determining how fast you need to travel, or figuring out the best deal on a purchase, these algebraic tools are essential for making informed decisions.
