Linear Equations & Inequalities Quiz, Explanations and Step-by-Step Review
Use the question set below to practice solving linear equations and solving linear inequalities in one variable: solve for x using inverse operations, simplify with the distributive property, combine like terms, solve equations with variables on both sides, and find a correct solution set for inequalities. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Jawab rangkaian soal dan tinjau kesalahanmu di akhir.
How this linear equations and inequalities practice works
1. Take the practice set: answer the linear equation and linear inequality questions below.
2. Open the lesson (optional): review solving steps, common mistakes, and quick checks.
3. Retry: return to the question set and apply the method immediately to improve speed and accuracy.
What you will learn in the linear equations and inequalities lesson
Foundations & vocabulary
Variable, coefficient, constant (parts of a linear expression)
Linear equation: solve for the value that makes both sides equal
Solution check: substitute your answer to verify it works
One-step & two-step equations
Inverse operations: undo \(+/-\) and \(\times/\div\)
Two-step form \(ax+b=c\): subtract \(b\), then divide by \(a\)
Decimals & fractions: solve by multiplying to clear the coefficient
Multi-step equations
Distributive property: \(a(b+c)=ab+ac\)
Combine like terms before isolating the variable
Variables on both sides: one solution, no solution, or infinitely many solutions
Linear inequalities & solution sets
Inequality symbols: \( <, \le, >, \ge \)
Flip the sign when multiplying/dividing by a negative number
Interval notation and number line graphs to represent the solution set
Purpose: Build a clear, reliable method for solving linear equations and solving linear inequalities in one variable. You’ll practice isolating the variable, simplifying expressions, and writing correct solution sets.
Success criteria
Identify the parts of a linear expression: variable, coefficient, and constant.
Solve one-step equations using inverse operations (undo \(+/-\) or \(\times/\div\)).
Solve two-step equations of the form \(ax+b=c\).
Solve multi-step linear equations with parentheses by using the distributive property and combining like terms.
Solve equations with variables on both sides and recognize one solution, no solution, or infinitely many solutions.
Solve linear inequalities and remember to reverse the inequality sign when multiplying or dividing by a negative number.
Write a solution set using inequality notation and interval notation (for example, \(x<3\) or \((-\infty,3)\)).
Check an equation solution by substituting back into the original equation.
Key vocabulary
Variable: a symbol (often \(x\)) that stands for a number.
Coefficient: the number multiplying the variable (in \(5x\), the coefficient is \(5\)).
Constant: a number without a variable term.
Linear equation: an equation that can be written as \(ax+b=c\) (one variable, exponent 1).
Solution: a value that makes the equation true.
Inverse operations: operations that undo each other (add/subtract, multiply/divide).
Distributive property: \(a(b+c)=ab+ac\).
Like terms: terms with the same variable power (e.g., \(3x\) and \(-2x\)).
Linear inequality: an inequality like \(ax+b<c\) with a set of solutions.
Solution set: all numbers that make an inequality true.
Quick pre-check
Pre-check 1: Solve for \(x\): \(-2x = 6\).
Hint: Divide both sides by \(-2\).
Pre-check 2: What happens to an inequality when you multiply both sides by a negative number?
Hint: This is the biggest difference between solving equations and solving inequalities.
One-Step Equations
Solve one-step linear equations
Learning goal: Solve one-step equations by using inverse operations to isolate the variable.
Key idea
To solve a linear equation, your goal is to get the variable alone on one side. You do that by using inverse operations: addition undoes subtraction, and multiplication undoes division. Whatever you do to one side, you must do to the other side to keep the equation balanced.
Worked example
Example: Solve \(\dfrac{x}{4}=2\).
Multiply both sides by \(4\): \[ \frac{x}{4}\cdot 4 = 2\cdot 4 \quad \Rightarrow \quad x=8. \]Check: \(8/4=2\) ✓
Try it
Try it 1: Solve for \(x\): \(5 - x = 1\).
Hint: Subtract \(5\) from both sides, then multiply by \(-1\).
Try it 2: Solve the equation \(x - 8 = 2\). What is \(x\)?
Hint: Add \(8\) to both sides.
Summary
Use inverse operations to isolate the variable.
Keep the equation balanced by doing the same thing to both sides.
Two-Step Equations
Solve two-step linear equations
Learning goal: Solve equations like \(ax+b=c\) by undoing operations in reverse order.
Key idea
For an equation such as \(ax+b=c\), the variable \(x\) is affected by two operations: first it’s multiplied by \(a\), then \(b\) is added. To solve, do the inverse operations in reverse order: subtract \(b\), then divide by \(a\).
Worked example
Example: Solve \(3x - 4 = 5\).
Add \(4\) to both sides: \[ 3x - 4 + 4 = 5 + 4 \quad \Rightarrow \quad 3x = 9. \] Divide both sides by \(3\): \[ x = 3. \]Check: \(3(3)-4=9-4=5\) ✓
Try it
Try it 1: Solve \(5x + 5 = 0\). What is \(x\)?
Hint: Subtract \(5\), then divide by \(5\).
Try it 2: Solve \(0.2x + 1 = 3\). What is \(x\)?
Hint: Subtract \(1\). Then divide by \(0.2\) (same as multiplying by \(5\)).
Summary
Undo addition/subtraction first, then undo multiplication/division.
Decimals work the same way: isolate the variable carefully and keep the equation balanced.
Multi-Step Equations
Multi-step equations: parentheses and like terms
Learning goal: Solve multi-step linear equations using the distributive property and combining like terms.
Key idea
When an equation includes parentheses or many terms, simplify first: (1) use the distributive property to remove parentheses, (2) combine like terms, then (3) isolate the variable using inverse operations.
Worked example
Example: Solve \(5x - (2x + 1) = 2\).
Distribute the minus sign and combine like terms: \[ 5x - 2x - 1 = 2 \quad \Rightarrow \quad 3x - 1 = 2. \] Add \(1\) to both sides, then divide by \(3\): \[ 3x = 3 \quad \Rightarrow \quad x=1. \]Check: \(5(1)-(2(1)+1)=5-(2+1)=2\) ✓
Try it
Try it 1: Solve \(4(x + 1) = 12\). What is \(x\)?
Hint: Distribute \(4\): \(4x+4=12\).
Try it 2: Solve \(3(x + 2) = 12\). What is \(x\)?
Hint: Distribute \(3\): \(3x+6=12\).
Summary
Distribute first, then combine like terms, then solve.
Write neat steps to avoid sign mistakes.
Variables on Both Sides
Equations with variables on both sides
Learning goal: Solve equations with \(x\) on both sides and recognize one-solution, no-solution, and all-real-numbers cases.
Key idea
To solve equations with \(x\) on both sides, move all \(x\)-terms to one side and all constants to the other. After simplifying:
If you get \(x = \text{number}\), there is one solution.
If you get a false statement like \(1 = -2\), there is no solution.
If you get a true statement like \(0=0\), there are infinitely many solutions (all real numbers).
Worked example
Example: Solve \(3x - 4 = 2x + 6\).
Subtract \(2x\) from both sides: \[ x - 4 = 6 \] Add \(4\) to both sides: \[ x = 10. \]
Try it
Try it 1: Solve for \(x\): \(4 - x = 2x + 1\).
Hint: Move all \(x\)-terms to one side and constants to the other.
Try it 2: Solve: \(3x + 1 = 3x - 2\).
Hint: Subtract \(3x\) from both sides. If you get something false, there is no solution.
Summary
Move variable terms to one side, constants to the other.
Watch for special results: false statement (no solution) or true statement (all real numbers).
Inequalities Basics
Linear inequalities: solve and write the solution set
Learning goal: Solve basic linear inequalities and describe the solution set using inequality notation and interval notation.
Key idea
Solving a linear inequality is like solving an equation, but the answer is a set of values. You can still add or subtract the same number on both sides. The key rule: When you multiply or divide by a negative number, you must reverse the inequality sign.
Worked example
Example: Solve the inequality \(x + 2 < 5\).
Subtract \(2\) from both sides: \[ x < 3. \] Solution set in interval notation: \((-\infty,3)\). On a number line, use an open circle at \(3\) and shade to the left.
Try it
Try it 1: Solve: \(4x + 1 > 9\).
Hint: Subtract \(1\), then divide by \(4\).
Try it 2: Solve \(5 - 2x \le 1\). What is the solution set?
Hint: After you get \(-2x \le -4\), divide by \(-2\) and reverse the sign.
Summary
Inequalities have solution sets, not just one number.
Reverse the inequality sign only when multiplying or dividing by a negative number.
Multi-Step Inequalities
Multi-step inequalities and special cases
Learning goal: Solve multi-step inequalities, including negatives, and recognize when an inequality has no solution.
Key idea
Multi-step inequalities use the same skills as multi-step equations: distribute, combine like terms, and isolate the variable. The key rule still applies: if you multiply or divide by a negative number, reverse the inequality sign.
Worked example
Example: Solve \(-4x + 3 > 11\).
Subtract \(3\) from both sides: \[ -4x > 8. \] Divide by \(-4\) and reverse the sign: \[ x < -2. \]
Try it
Try it 1: Solve: \(3x - 4 < 2\).
Hint: Add \(4\), then divide by \(3\).
Try it 2: Solve: \(3x + 1 < 3x - 2\).
Hint: Subtract \(3x\) from both sides and decide whether the remaining statement is true or false.
Worked solution
Subtract \(3x\) from both sides: \[ 1 < -2. \] This is false, so there are no values of \(x\) that make the inequality true. The solution set is empty.
Summary
Reverse the inequality sign only when multiplying or dividing by a negative number.
If your work ends with a false statement, the inequality has no solution.
Applications & History
Why linear equations and inequalities matter
Learning goal: Connect solving linear equations and inequalities to real-life situations and build confidence with algebraic reasoning.
Science: formulas like \(d=vt\) (distance = speed × time) often lead to linear equations.
Planning: inequalities model limits and constraints (time, cost, capacity).
Everyday reasoning: “at least,” “no more than,” and “must be greater than” translate naturally into inequalities.
Worked example: a budget inequality
Example: A gym charges a $20 signup fee plus $15 per month. You want to spend no more than $80. How many months can you afford?
Write and solve an inequality: \[ 15m + 20 \le 80 \]\[ 15m \le 60 \quad \Rightarrow \quad m \le 4 \] You can afford up to 4 months.
Try it
Try it 1: A taxi charges $4 plus $2 per kilometer. You have $16. What is the greatest number of kilometers you can ride?
Hint: Model with \(2k+4 \le 16\), then solve for \(k\).
Common mistake check
Try it 2: Which step requires reversing the inequality sign?
Hint: Only one operation can change the direction of an inequality.
Fun facts (a little history)
Algebra roots: Systematic equation-solving methods were developed and recorded in early algebra, including work associated with al-Khwarizmi.
Inequality symbols: The symbols \(<\) and \(>\) became standard in mathematics to compare numbers and expressions, and they are now essential for describing solution sets.
Big idea: Linear equations model a constant rate of change, which is why they appear everywhere from finance to physics.
Final recap
Equations: isolate the variable using inverse operations and check by substitution.
Multi-step equations: distribute, combine like terms, then solve.
Variables on both sides: watch for one solution, no solution, or all real numbers.
Inequalities: solve like equations, but reverse the sign when multiplying/dividing by a negative.
Solution sets: describe with inequality notation (like \(x\ge 2\)) or interval notation (like \([2,\infty)\)).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the solving skill you need (one-step, two-step, multi-step, or inequalities).
