Logic in Algebra
A free Logic lesson from the “Logic Applications and Final Review” unit, with a worked example and practice problems including step-by-step solutions.
Algebra uses logic whenever students decide which operations are allowed, what a solution set means, or whether a rule applies to every case. Learning objective: Use logic to read algebra rules, solution sets, and conditional claims. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: An algebra rule may apply only if a denominator is not zero. Example 2: A program condition such as 'if score >= 70 and quiz submitted' uses logic to decide what happens next. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Use logic to read algebra rules, solution sets, and conditional claims
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Logic in Algebra): Worked example: Is x = 3 a solution of 2x + 1 = 7?
- Substitute the value into the equation.
- Replace x with 3 and simplify each side.
- Both sides match, so it is a solution.
Answer: Yes
Lesson preview: this page shows 5 practice items. The interactive activity contains 20 practice items and 10 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A (Logic in Algebra): Practice: Is x = 3 a solution of 2x + 1 = 7?
Choices: Yes · No
Show solution
- Substitute the value into the equation.
- Replace x with 3 and simplify each side.
- Both sides match, so it is a solution.
Answer: Yes
2. Practice case B (Logic in Algebra): Practice: "If a value is a solution of an equation, then substituting it makes the two sides..." Complete the conditional.
Choices: equal · unequal · both zero · both positive
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- A solution satisfies the equation.
- Substituting it makes left side = right side.
- So the two sides are equal.
Answer: equal
3. Practice case C (Logic in Algebra): Practice: A solution set is x ≤ 2. Is x = 2 in the solution set?
Choices: Yes · No
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- Check whether 2 satisfies x ≤ 2.
- The symbol includes the endpoint.
- So the value is included.
Answer: Yes
4. Practice case D (Logic in Algebra): Practice: To undo "add 4" on one side of an equation, you should:
Choices: subtract 4 · multiply by 4 · divide by 4 · add 4 again
Show solution
- Inverse operations undo each other.
- The inverse of "add 4" is "subtract 4".
- Apply it to both sides to keep the equation balanced.
Answer: subtract 4
5. Practice case E (Logic in Algebra): Practice: Why can squaring both sides of an equation require checking the result?
Choices: It can introduce an extraneous solution. · It always makes every solution invalid. · It removes the need for conditions. · It proves both sides were positive.
Show solution
- Squaring is not reversible for negatives.
- It can create values that fail the original equation.
- So you must check candidates in the original.
Answer: It can introduce an extraneous solution.
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