L3 · Linear inequalities: what a solution set means

Learning goal

“I can explain what the solution set of a linear inequality is - an infinite range of values, not a single number - test whether a given value is a solution, and solve simple linear inequalities using the addition and positive-multiplication properties, verifying with a member and a non-member.”

Inequality — a statement comparing two expressions with <, >, ≤, or ≥, like 3x - 2 < 20 (review from Grade 8).

Solution set — the set of ALL values that make the inequality true. For an equation it was a list you could finish; for an inequality it is a range you describe.

Satisfies — a value satisfies an inequality if substituting it makes the statement true: 6 satisfies 3x - 2 < 20 because 16 < 20.

Boundary value — the value where the two sides would be EQUAL (for 3x ≥ 12, the boundary is x = 4, because 3(4) = 12). Whether the boundary itself counts as a solution depends on the symbol: ≤ and ≥ include it; < and > exclude it.

Worked example

Which of -2, 0, 3, 6 satisfy 2x - 1 ≥ 5? Test each by substitution: -2: 2(-2) - 1 = -5. Is -5 ≥ 5? No. 0: 2(0) - 1 = -1. Is -1 ≥ 5? No. 3: 2(3) - 1 = 5. Is 5 ≥ 5? YES - ≥ includes equality. 6: 2(6) - 1 = 11. Is 11 ≥ 5? Yes. Say it plainly: an equation's solution was a value you could point at. An inequality's solution is a whole crowd - every value on one side of a boundary. You never list them all; you describe the set, and you test membership one value at a time by substitution.

Check yourself — explain it

3 satisfies 2x - 1 ≥ 5 even though it makes the two sides equal, because...

Paired problem — example still visible

Is 0 a solution of 5 - 2x > 1? -> Yes: 5 - 2(0) = 5, and 5 > 1 (and note the strict symbol: equality would NOT have counted)

Faded completion — last step is yours

"Test x = 2 in x - 7 < -4: 2 - 7 = -5, and the statement -5 < -4 is ___ (true/false), so 2 ___ (is/is not) a solution." -> true; is

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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