A community library is arranging a new-releases display shelf near the entrance that can hold a number of books \(x\) such that \(-3 \lt 2x+1 \leq 9\). Solv...
Question 1 Report
A community library is arranging a new-releases display shelf near the entrance that can hold a number of books \(x\) such that \(-3 \lt 2x+1 \leq 9\). Solve this inequality and list all the integer values of \(x\) that satisfy it, showing them on a number line.
(4)
Answer Details
A double inequality like \(-3\lt2x+1\leqslant9\) is really two separate inequalities sharing the middle expression; each is solved the same way as a single inequality, and their two solutions are then combined.
Taking the left-hand inequality: \(-3\lt2x+1\). Subtracting 1: \(-4\lt2x\). Dividing by 2: \(-2\lt x\). [1 mark]
Taking the right-hand inequality: \(2x+1\leqslant9\). Subtracting 1: \(2x\leqslant8\). Dividing by 2: \(x\leqslant4\). [1 mark]
Combining both parts: \(-2\lt x\leqslant4\). The integer values satisfying this are \(-1, 0, 1, 2, 3, 4\). [1 mark]
This is shown on the number line above with an open circle at \(-2\) (not included), a closed circle at \(4\) (included), and the region between them shaded. [1 mark]
Exam tip: an open circle always means "up to but not including" (used for \(\lt\) or \(\gt\)), while a filled circle means "including" (used for \(\leqslant\) or \(\geqslant\)).
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