L1 · Strict and non-strict inequalities; number-line rays

Learning goal

“I can tell whether an inequality is strict or non-strict, and I can model any inequality with <, >, ≤ or ≥ on a number line - the right dot, at the right number, shading the right way.”

Inequality — a comparison of two numbers or expressions using <, >, ≤ or ≥. Unlike an equation, it usually has a whole range of solutions, not one value.

Strict inequality — uses < or >. The boundary value itself does NOT make it true: 3 > 3 is false.

Non-strict inequality — uses ≤ or ≥. The boundary value DOES make it true, because "or equal to" is included: 3 ≥ 3 is true.

Ray — the number-line picture of an inequality: a dot at the boundary value with shading running off in one direction, forever.

Worked example

Compare x > 3 and x ≥ 3 by testing the boundary, 3 itself. Step 1: Substitute 3 into x > 3: is 3 > 3 true? No - a number is never greater than itself. So 3 is not a solution of x > 3. The inequality is strict. Step 2: Substitute 3 into x ≥ 3: is 3 ≥ 3 true? Yes - ≥ means "greater than or equal to," and 3 is equal to 3. So 3 is a solution of x ≥ 3. The inequality is non-strict. Step 3: Test a value past the boundary, 4: 4 > 3 true, 4 ≥ 3 true - both work. The two inequalities differ at exactly one point: the boundary itself. Say it plainly: the whole strict/non-strict question is settled by one test - does the boundary value itself work?

Check yourself — explain it

x ≤ 10 is non-strict because...

Paired problem — example still visible

Classify x < -2 as strict or non-strict, and test the boundary to prove it. -> Strict: -2 < -2 is false, so -2 is not a solution.

Faded completion — last step is yours

"For x ≥ -5: substitute the boundary -5: -5 ≥ -5 is ______ (true/false), so -5 ______ (is / is not) a solution, and the inequality is ______." -> true; is; non-strict

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