L4 · Graphing inequality solutions; set-builder notation

Learning goal

“I can represent the solution set of a linear inequality three ways - as a number-line graph described in words, as an inequality, and in set-builder notation - and I can solve inequalities with negative coefficients, reversing the symbol when multiplying or dividing by a negative, and prove the reversal was needed by testing values.”

Open circle — the hollow dot on a number-line graph marking a boundary that is NOT included (used for < and >).

Closed circle — the filled dot marking a boundary that IS included (used for ≤ and ≥).

Set-builder notation — the compact set description {x | x ≤ -2, x ∈ R}, read "the set of all x such that x is less than or equal to -2, with x a real number" (built in G9U01, now applied to solution sets).

Ray — the arrowed half-line on the graph: from the boundary dot, it runs in the direction of the values that satisfy the inequality.

Worked example

Graph x ≥ -2 (described in words). Dot: the symbol ≥ includes its boundary, so a CLOSED circle at -2. Ray: the solutions are -2 and everything greater, so the arrow runs RIGHT from -2, and the line from -2 rightward is shaded. Say it plainly: the symbol tells you both facts. Its "equals bar" (or its absence) chooses the dot; its direction, matched against the values that work, chooses the arrow. Read every graph as a sentence: dot type, position, direction.

Check yourself — explain it

The graph of x > 5 uses an open circle because...

Paired problem — example still visible

Describe the graph of x > 5 in words -> Open circle at 5; arrow (ray) running to the right

Faded completion — last step is yours

"Graph of x ≤ 0: a ___ circle at 0, with the arrow running to the ___" -> closed; left

Ready to practise?

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

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