English:Linear Equations and Inequalities
Linear Equations and Inequalities
Introduction
Linear Equations and Inequalities is an aiMOOC for Grades 9–10. You will learn how to represent unknown quantities, solve linear equations and inequalities, graph their solutions, interpret slope and intercepts, and build mathematical models from real situations. The central idea is balance: an equation stays true when you perform the same valid operation on both sides, while an inequality requires extra care because multiplying or dividing by a negative number reverses the order.
Linear mathematics appears whenever one quantity changes at a constant rate. Examples include wages earned per hour, distance traveled at constant speed, temperatures converted between scales, fixed fees plus usage charges, and limits such as budgets or capacity. These ideas connect Algebra, linear functions, coordinate systems, Slope, number lines, and mathematical modeling.
Learning Goals
By the end of this course, you should be able to explain what makes an equation or inequality linear, solve one-variable linear equations, check solutions by substitution, solve one-variable linear inequalities, graph solution sets on a number line, interpret and graph linear equations in two variables, graph linear inequalities in two variables, and translate real-world constraints into algebraic statements.
You should also be able to communicate your reasoning clearly. A correct answer matters, but so does showing why each transformation preserves equality or correctly changes an inequality.
Foundations of Linear Algebra
Expressions, Equations, and Inequalities
An expression combines numbers, variables, and operations, such as 4x - 7. An equation states that two expressions are equal, such as 4x - 7 = 13. An inequality compares two expressions using symbols such as <, >, ≤, or ≥, for example 4x - 7 ≥ 13.
A solution is a value that makes a statement true. For an equation, the solution may be one value, no value, or infinitely many values. For an inequality, the solution is usually a set or interval of values.
Important vocabulary includes the variable, the coefficient multiplying a variable, and a constant term. In 5x + 9, x is the variable, 5 is the coefficient, and 9 is the constant.
The balance image is useful because an equation behaves like a balanced scale. If you change only one side, the equality usually becomes false. If you apply the same reversible operation to both sides, the equality is preserved.
What Makes a Relationship Linear?
A one-variable equation such as 3x + 5 = 20 is linear because the variable has exponent 1 and is not multiplied by another variable. In two variables, a common form is ax + by = c, where a and b are not both zero. Its graph is a straight line.
A linear function is often written as y = mx + b. Here m is the slope and b is the y-intercept. The slope is the constant rate of change: for every increase of 1 in x, y changes by m.
Nonlinear examples include x² + 1 = 10, xy = 12, and y = 1/x. They do not have a constant rate of change over their entire domains.
Solving Linear Equations
The Balance Principle
To solve an equation, your goal is to isolate the variable while preserving equality. You may add, subtract, multiply, or divide both sides by the same number, except that division by zero is never allowed.
For example, solve 3x + 5 = 20.
- Subtract 5 from both sides: 3x = 15.
- Divide both sides by 3: x = 5.
- Check: 3 · 5 + 5 = 20, so the solution is correct.
The check is important because it catches arithmetic and sign errors.
Variables on Both Sides
Consider 5x - 7 = 2x + 8. First collect the variable terms on one side and constants on the other.
- Subtract 2x from both sides: 3x - 7 = 8.
- Add 7 to both sides: 3x = 15.
- Divide by 3: x = 5.
Sometimes the variable cancels completely. If you simplify an equation and obtain a true statement such as 4 = 4, every value in the domain is a solution. If you obtain a false statement such as 4 = 9, there is no solution.
Parentheses and Fractions
Use the distributive property before combining like terms. For example, 2(x + 3) = 14 becomes 2x + 6 = 14, then 2x = 8, so x = 4.
Fractions can often be simplified by multiplying every term by a common denominator. For example, x/3 + 2 = 5 becomes x + 6 = 15 after multiplying both sides by 3, so x = 9.
When clearing fractions, multiply every term on both sides by the common denominator. Missing one term changes the equation.
Graphing Linear Equations
Slope and Intercepts
In y = mx + b, the slope m measures vertical change divided by horizontal change. The y-intercept b is the y-value where the line crosses the y-axis.
If y = -2x + 6, then the slope is -2 and the y-intercept is 6. Starting at (0, 6), move right 1 and down 2 to locate another point. Repeating that movement traces the line.
Graphing from an Equation
You can graph a line in several ways. One method is to identify the y-intercept and use the slope. Another is to make a table of x-values and calculate the corresponding y-values. A third is to find the x- and y-intercepts.
For y = 2x + 6, the line crosses the y-axis at 6 and rises 2 units for each 1-unit move to the right. Any point on the line satisfies the equation, and any point not on the line does not.
Parallel, Perpendicular, and Intersecting Lines
Parallel nonvertical lines have the same slope and different y-intercepts. Perpendicular nonvertical lines have slopes whose product is -1, so their slopes are negative reciprocals. Intersecting lines have one common point, which can represent a shared solution when two linear equations are considered together.
These relationships help you reason about systems of linear equations, geometric design, and coordinate proofs.
Solving Linear Inequalities
Inequalities Describe Ranges
An equation such as x = 4 identifies a specific value. An inequality such as x > 4 describes all values greater than 4. The solution set can be represented using words, inequality notation, interval notation, or a graph on a number line.
Use an open circle for a strict inequality such as x > 4 or x < 4 because the endpoint is excluded. Use a closed circle for x ≥ 4 or x ≤ 4 because the endpoint is included.
The Sign-Reversal Rule
Most equation-solving moves also work for inequalities. However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
For example, solve -3x > 12. Dividing both sides by -3 gives x < -4.
Why does the sign reverse? On the number line, multiplying by -1 reflects every number across zero. If 2 < 5, then -2 > -5 after reflection. The order reverses.
Compound Inequalities
A compound inequality combines conditions. The statement -2 < x ≤ 5 means x is greater than -2 and at most 5. Both conditions must be true.
The statement x < -3 or x > 4 describes two separate regions. In an and statement, you usually look for overlap. In an or statement, you usually combine the allowed regions.
When modeling a real situation, words such as at least often translate to ≥, at most to ≤, more than to >, and less than to <.
Linear Inequalities in Two Variables
Boundary Lines and Half-Planes
A two-variable linear inequality such as y > 2x - 1 has infinitely many ordered-pair solutions. Its graph is a region of the coordinate plane.
First graph the related boundary line y = 2x - 1. Use a dashed boundary for < or > because points on the boundary are not included. Use a solid boundary for ≤ or ≥ because boundary points are included. Then test a point not on the boundary, often (0, 0), to decide which side to shade.
A shaded half-plane represents all ordered pairs that make the inequality true. Always check a sample point from your shaded region by substituting its coordinates into the original inequality.
Modeling Real Situations
From Words to Equations
Suppose a bicycle rental costs a fixed fee of $8 plus $4 per hour. If the total cost is C and the number of hours is h, then C = 4h + 8. The slope 4 represents the hourly rate, and the intercept 8 represents the starting fee.
If you paid $28, solve 28 = 4h + 8. Subtract 8 to get 20 = 4h, then divide by 4 to get h = 5 hours.
A strong model identifies what each variable means, includes units, and explains what slope and intercept mean in context.
From Limits to Inequalities
Suppose you have a budget of $50 for notebooks that cost $6 each after a fixed delivery fee of $8. If n is the number of notebooks, then 6n + 8 ≤ 50. Solving gives 6n ≤ 42, so n ≤ 7. Because n counts notebooks, only nonnegative whole-number solutions make sense.
Context can restrict the mathematical solution set. Algebra may produce every real number below 7, but the situation may allow only 0, 1, 2, 3, 4, 5, 6, or 7 notebooks.
Reasonableness and Error Checking
When you finish a problem, ask whether your result fits the context. A negative number of tickets, a fractional person, or a cost above a stated budget may signal that you misinterpreted the model.
Common errors include distributing incorrectly, combining unlike terms, changing only one side of an equation, forgetting to reverse an inequality after dividing by a negative number, using a solid line for a strict two-variable inequality, and shading the wrong half-plane.
A reliable routine is to simplify carefully, show equivalent steps, check the final result in the original statement, and explain the answer in words with appropriate units.
Interactive Tasks
Quiz: Test Your Knowledge
Solve 3x + 5 = 20. (x = 5) (!x = 15) (!x = 25) (!x = 45)
Solve 5x - 7 = 2x + 8. (x = 5) (!x = 1) (!x = 3) (!x = 15)
Which action preserves an equation when it is applied correctly? (Performing the same operation on both sides) (!Adding a number to only the left side) (!Changing a coefficient without changing the other side) (!Dividing only one side by a nonzero number)
What is the slope of y = -2x + 7? (-2) (!2) (!7) (!-7)
What is the y-intercept of y = 3x - 4? (-4) (!3) (!4) (!-3)
Solve -3x > 12. (x < -4) (!x > -4) (!x < 4) (!x > 4)
When must you reverse an inequality sign while solving? (When multiplying or dividing by a negative number) (!When adding a positive number) (!When subtracting the same number from both sides) (!When combining like terms)
Solve 2x + 3 ≤ 11. (x ≤ 4) (!x ≥ 4) (!x ≤ 7) (!x ≥ 7)
Which boundary should be used when graphing y > 2x - 1? (A dashed line) (!A solid line) (!A vertical line only) (!No boundary line)
What feature identifies a linear relationship in a graph? (A constant rate of change) (!A changing slope at every point) (!A curved path) (!A variable raised to the second power)
Memory Game
| Variable | Symbol representing a quantity that can change |
| Coefficient | Number multiplying a variable |
| Constant | Term with no variable |
| Slope | Vertical change divided by horizontal change |
| Intercept | Point where a graph crosses an axis |
| Solution set | All values that make a statement true |
| Boundary line | Line separating solution and nonsolution regions |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Additive inverse | Operation used to remove an added constant |
| Distributive property | Rule used to expand a factor across parentheses |
| Negative multiplier | Situation that reverses an inequality sign |
| Closed endpoint | Number-line mark showing an included boundary value |
| Test point | Coordinate used to choose the shaded half-plane |
...
Crossword Puzzle
| Coefficient | What do you call the number multiplying a variable? |
| Variable | What symbol stands for a quantity that may change? |
| Slope | What measures the constant rate of change of a line? |
| Intercept | What is the point where a graph crosses an axis called? |
| Inequality | What mathematical statement compares two expressions using order symbols? |
| Solution | What do you call a value that makes an equation or inequality true? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Balance model: Build a physical or digital balance model that shows why doing the same operation to both sides preserves an equation. Include one solved example and a short explanation.
- Equation card sort: Create a set of cards showing steps from three linear equations, mix the steps, and challenge a partner to put each solution process in a valid order.
- Number-line poster: Design a poster that compares strict and inclusive inequalities using open and closed endpoints, arrows, and at least four correctly labeled examples.
- Linear graph sketch: Draw three lines with positive, negative, and zero slope. Label each slope and y-intercept and explain how you can see the rate of change.
Standard
- Phone plan model: Compare two fictional phone plans with linear cost equations. Make a table and graph, then explain when each plan is cheaper.
- Temperature inequality: Create and solve an inequality describing a safe temperature range for a realistic situation. Represent the solution on a number line and interpret it in words.
- Mini-lesson video: Produce a two- to four-minute video teaching how to solve a linear inequality that requires division by a negative number. Include a visual explanation of the sign reversal.
- Interview with a trade professional: Interview someone in a trade, technical field, business, or service job about a situation involving rates, costs, limits, or measurements. Translate one example into a linear equation or inequality.
Advanced
- Optimization investigation: Design a small budget or capacity problem with at least two constraints. Represent the constraints as linear inequalities and justify which solutions are realistic.
- Data-fitting project: Collect paired data that is approximately linear, create a scatter plot, estimate a line of best fit, and interpret the slope and intercept in context.
- Inequality design challenge: Create a coordinate-plane design formed by at least five shaded linear inequalities. Provide the inequalities and explain how each boundary controls part of the design.
- Error-analysis portfolio: Collect or invent six incorrect solutions to linear equations and inequalities. Diagnose each error, correct it, and write a general rule that would prevent the mistake.
Learning Assessment
- Equation reasoning assessment: Solve a multi-step equation with parentheses and fractions, justify each transformation, and verify the result by substitution.
- Inequality transfer assessment: Solve an inequality containing a negative coefficient, explain why the sign changes, and represent the final solution on a number line.
- Graph interpretation assessment: Analyze a linear graph, determine its slope and intercept, write an equation for it, and explain what those parameters could mean in a real situation.
- Modeling assessment: Translate a realistic fixed-fee-plus-rate situation into a linear equation, solve a question from the model, and evaluate whether the result is reasonable.
- Constraint assessment: Translate a budget or capacity limit into a linear inequality, solve it, and explain how the context restricts the mathematical solution set.
- Two-variable inequality assessment: Graph a linear inequality in two variables, choose the correct boundary style, shade the appropriate half-plane, and verify a sample point.
Evidence of Learning
Knowledge: You can distinguish expressions, equations, inequalities, and linear relationships; explain slope, intercepts, solution sets, and boundary lines; and state the sign-reversal rule accurately.
Skills: You can solve and check linear equations and inequalities, graph one- and two-variable solution sets, interpret slope and intercepts, translate words into algebra, and justify transformations logically.
Products: Strong evidence may include a correct graphing portfolio, a modeling report, a short teaching video, a number-line visualization, a data-based linear model, or a designed system of inequalities.
Transfer: You can recognize constant-rate relationships and real constraints outside textbook exercises, choose an equation or inequality appropriately, interpret the result in context, and detect unreasonable answers.
OERs on the Topic
The following English Wikipedia articles provide open reference material for further study.
Linked Learning Areas