Question
3. Solve the inequality 4x + 2 ≤ 18 and justify each step using algebraic properties or properties of real numbers.
Explain how these properties ensure that the solution is valid for all real numbers that satisfy the inequality
Explain how these properties ensure that the solution is valid for all real numbers that satisfy the inequality
Answers
GPT-5 mini
Start with the inequality:
4x + 2 ≤ 18. (given)
1) Subtract 2 from both sides:
4x + 2 − 2 ≤ 18 − 2 (Addition property of inequality: for any real c, if a ≤ b then a + c ≤ b + c; here c = −2)
⇒ 4x ≤ 16.
2) Divide both sides by 4:
(4x)/4 ≤ 16/4 (Division / multiplication property of inequality: if a ≤ b and c > 0, then a/c ≤ b/c; here c = 4 > 0)
⇒ x ≤ 4.
So the solution set is {x ∈ R | x ≤ 4}, or in interval notation (−∞, 4].
Why these steps are valid for all real numbers satisfying the inequality:
- The addition property of inequalities and the multiplication/division property for positive scalars are consequences of the order axioms of the real numbers; they hold for every real a, b and any real c (or c > 0 for multiplication/division).
- Closure of the real numbers under addition and multiplication guarantees the expressions obtained at each step are real.
- Because we only added the same number to both sides and divided by a positive number, we preserved the truth of the inequality in both directions; reversing the steps (multiply by 4, then add 2) shows every x ≤ 4 indeed satisfies 4x + 2 ≤ 18, so the solution set is exact.
(If we had divided by a negative number we would have to reverse the inequality sign; that is not needed here because 4 > 0.)
4x + 2 ≤ 18. (given)
1) Subtract 2 from both sides:
4x + 2 − 2 ≤ 18 − 2 (Addition property of inequality: for any real c, if a ≤ b then a + c ≤ b + c; here c = −2)
⇒ 4x ≤ 16.
2) Divide both sides by 4:
(4x)/4 ≤ 16/4 (Division / multiplication property of inequality: if a ≤ b and c > 0, then a/c ≤ b/c; here c = 4 > 0)
⇒ x ≤ 4.
So the solution set is {x ∈ R | x ≤ 4}, or in interval notation (−∞, 4].
Why these steps are valid for all real numbers satisfying the inequality:
- The addition property of inequalities and the multiplication/division property for positive scalars are consequences of the order axioms of the real numbers; they hold for every real a, b and any real c (or c > 0 for multiplication/division).
- Closure of the real numbers under addition and multiplication guarantees the expressions obtained at each step are real.
- Because we only added the same number to both sides and divided by a positive number, we preserved the truth of the inequality in both directions; reversing the steps (multiply by 4, then add 2) shows every x ≤ 4 indeed satisfies 4x + 2 ≤ 18, so the solution set is exact.
(If we had divided by a negative number we would have to reverse the inequality sign; that is not needed here because 4 > 0.)