L2 · Verifying a possible solution by substitution

Learning goal

“I can test any value in an inequality - substituting carefully, evaluating both sides, and letting the true/false result decide - including boundary values, negatives, fractions and decimals.”

Solution (of an inequality) — any value of the variable that makes the inequality a true statement. There are usually infinitely many.

Substitute — replace the variable with a chosen value, keeping every operation and sign exactly as written.

Verify — carry out the substitution and the comparison, and state the verdict: solution or not a solution.

Boundary value — the number the variable is compared with; testing it separates strict from non-strict behaviour.

Worked example

Is 5 a solution of x + 4 ≤ 9? Step 1 (substitute): replace x with 5: 5 + 4 ≤ 9. Step 2 (evaluate): 5 + 4 = 9, so the statement is 9 ≤ 9. Step 3 (compare): ≤ means "less than or equal to" - 9 is equal to 9, so the statement is TRUE. Step 4 (verdict): 5 is a solution. Notice: 5 is exactly the boundary case, and it passes because the inequality is non-strict.

Check yourself — explain it

In the check 9 ≤ 9, the verdict is "true" because...

Paired problem — example still visible

Is 6 a solution of 2x - 1 < 10? -> No: 2(6) - 1 = 11, and 11 < 10 is false.

Faded completion — last step is yours

"Is 1/2 a solution of x - 3 ≥ -3? Substitute: 1/2 - 3 = ______. Compare with -3: the statement is ______, so 1/2 ______ a solution." -> -2.5; true (-2.5 ≥ -3); is

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