Why must you check whether a diagram shades the wanted or the unwanted region?

Because the two conventions give opposite inequality signs for the same picture. If the shading marks the unwanted side, the region being described is the unshaded one, so every sign flips compared with reading the shaded side.

IGCSEMathsEdexcelMaths BFlashcardsFunctionsGraphing Inequalities

Graphing Inequalities (Edexcel IGCSE Maths B): Flashcards

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FrontRepresenting Inequalities as Regions
What does the solution to a 2D inequality such as $x + y \geq 8$ look like?

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Cards in this collection (12)

What does the solution to a 2D inequality such as $x + y \geq 8$ look like?
It is a region in the $x y$ plane, rather than a set of separate values.
An inequality in one variable such as $y \geq 2$ also gives a region, namely every point on or above the line $y = 2$ .
Fill in the two missing words about drawing boundary lines.
Use a $______$ line for $\leq$ or $\geq$ because the line is included, and a $______$ line for $<$ or $>$ because it is not.
The completed sentence is:
Use a solid line for $\leq$ or $\geq$ because the line is included, and a dotted line for $<$ or $>$ because it is not.
The style of the line is what tells a reader whether the boundary itself belongs to the region.
For $y < - 2 x + 5$ , which side of the line is the wanted region?
Below the line, because the inequality reads $y$ less than something.
If you are unsure, substitute a point from one side into the inequality: a true statement means that side is the wanted one.
True or False?
When showing a region $R$ defined by several inequalities, you shade the side of each line that you want.
False.
Unless the question says otherwise, shade the unwanted side of each line.
Shading away everything you do not want leaves the wanted region clear and unshaded, which is far easier to read than hunting for the one area covered by every layer of shading.
A region is defined by $y \geq x$ , $x + y \leq 8$ and $y \geq 2$ . Does the point $(3, 4)$ lie in it?
Yes, because it satisfies all three: $4 \geq 3$ is true, $3 + 4 = 7 \leq 8$ is true, and $4 \geq 2$ is true.
A point has to satisfy every inequality to lie in the region, so a single failure is enough to rule it out.
Why rearrange $3 x + 2 y \geq 12$ before drawing its boundary line?
Rearranging it to $y \geq - \frac{3}{2} x + 6$ puts it in the form $y = m x + c$ , so the gradient and the $y$ -intercept can be read straight off.
It also makes the wanted side obvious, because the inequality now reads $y$ greater than or equal to something.
What is the first thing to do when reading inequalities off a shaded region?
Find the equation of each line on the graph, ignoring the inequality signs for the moment.
Sloping lines need $y = m x + c$ from the gradient and $y$ -intercept, vertical lines have the form $x = k$ , and horizontal lines the form $y = k$ .
A boundary line crosses the $y$ -axis at 7 and falls one unit for every unit to the right. Fill in the two missing values.
$y =______x +______$
The completed equation is:
$y = - x + 7$
The gradient is $- 1$ because the line falls as it goes to the right, and the $y$ -intercept is the 7.
True or False?
A horizontal boundary line on a region diagram has an equation of the form $y = k$ .
True.
Every point on a horizontal line has the same $y$ -coordinate, so its equation sets $y$ equal to a constant.
It is the vertical line that has equation $x = k$ , and these two are the pair most often swapped.
A shaded region lies above the line $y = x$ , so which inequality does it give?
It gives $y > x$ if the line is dotted, or $y \geq x$ if the line is solid.
A region above a line gives $\geq$ or $>$ , and a region below it gives $\leq$ or $<$ .
A shaded region lies to the right of the solid line $x = 1$ . Which inequality is that?
It is $x \geq 1$ , because a region to the right of a vertical line gives $\geq$ or $>$ .
A region to the left of $x = k$ gives $\leq$ or $<$ instead.
Why must you check whether a diagram shades the wanted or the unwanted region?
Because the two conventions give opposite inequality signs for the same picture.
If the shading marks the unwanted side, the region being described is the unshaded one, so every sign flips compared with reading the shaded side.

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