Flashcards · Maths · Further Algebra
Inequalities and Regions
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What does a solid line mean on a graph of an inequality?
The line is included, so the inequality is \(\leq\) or \(\geq\).
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What does a dashed line mean?
The line is not included, so the inequality is \(<\) or \(>\).
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How do you decide which side of a line to shade?
Test a point, such as the origin, in the inequality.
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What does \(x \geq 1\) look like on a graph?
The region to the right of the vertical line \(x = 1\), including the line.
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What does \(y \leq 2x\) look like on a graph?
The region on or below the line \(y = 2x\).
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How do you describe a region bounded by three lines?
Write one inequality for each boundary.
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What are the integers satisfying \(-2 < x \leq 3\)?
\(-1, 0, 1, 2, 3\).
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Do points on a dashed line count as inside the region?
No.
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How many integer points satisfy \(x \geq 1\), \(y \geq 1\), \(x + y \leq 6\)?
15.
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What does the origin test show for \(x + y \leq 6\)?
\(0 + 0 \leq 6\) is true, so shade the origin side.
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How do you solve a quadratic inequality (Higher tier)?
Find the roots and use a sketch of the graph.
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Solve \(x^2 - x - 6 < 0\) (Higher tier).
\(-2 < x < 3\).
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Solve \(x^2 > 9\) (Higher tier).
\(x < -3\) or \(x > 3\).
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What does R usually stand for in these questions?
The region that satisfies all the inequalities.
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Where does the line \(x + y = 6\) cross the axes?
\((6, 0)\) and \((0, 6)\).
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Why check with a point inside the region?
All the inequalities must be true there.