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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

Why it matters: Logic transfers to algebra, geometry, statistics, computer science, AI prompts, and clear written explanations.

Worked example

Problem. Example case A (Logic in Algebra): Worked example: Is x = 3 a solution of 2x + 1 = 7?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Logic in Algebra): Worked example: Is x = 3 a solution of 2x + 1 = 7?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. Substitute the value into the equation.
  4. Replace x with 3 and simplify each side.

Answer: Yes

Practice problems

1. Practice case A (Logic in Algebra): Practice: Is x = 3 a solution of 2x + 1 = 7?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Logic in Algebra): Practice: Is x = 3 a solution of 2x + 1 = 7?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. Substitute the value into the equation.
  4. Replace x with 3 and simplify each side.
  5. Both sides match, so it is a solution.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: 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.
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. A solution satisfies the equation.
  4. Substituting it makes left side = right side.
  5. So the two sides are equal.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Logic in Algebra): Practice: A solution set is x ≤ 2. Is x = 2 in the solution set?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. Check whether 2 satisfies x ≤ 2.
  4. The symbol includes the endpoint.
  5. So the value is included.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

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
  1. Inverse operations undo each other.
  2. The inverse of "add 4" is "subtract 4".
  3. 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
  1. Squaring is not reversible for negatives.
  2. It can create values that fail the original equation.
  3. So you must check candidates in the original.

Answer: It can introduce an extraneous solution.

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