Representing-Inequalities-As-Regions Wjec-Eduqas Higher

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Exam code:C300

Finding regions using inequalities

What are 2D inequalities?

Recall that an inequality in one variable (1D inequality) represents a relationship that is not equal
- An inequality of x < 7, represents all values smaller than 7
  - There are an infinite number of values than can satisfy this inequality
A 2D inequality represents a relationship between two expressions that is not equal
- The inequality y > x represents all pairs of numbers x and y where the y value is greater than the x value
  - There are an infinite number of pairs of values that would satisfy this inequality
  - These pairs of numbers can be thought of as coordinates
  - On a graph, all coordinates above the line y = x would satisfy this inequality
If a 2D inequality includes either the symbol ≤ or ≥, then coordinates on the line itself also satisfies the inequality
- E.g. y ≤ 2x represents all of the pairs of numbers where the value of y is less than two lots of the value of x
  - This is the region below the line y = 2x, but also being on the line y = 2x satisfies the inequality

How do we draw inequalities on a graph?

A set of 2D inequalities can be shown graphically using straight lines and shaded regions
To draw the correct lines:
- Replace the inequality sign with “=” and draw that line
  - Use a solid line for ≤ or ≥ (to indicate the line is included)
  - Use dotted line for < or > (to indicate the line is not included)
To decide which side of the line is the wanted side:
- if “y ≤ …” or “y < …” then the wanted region is below the line
- if “y ≥ …” or “y > …” then the wanted region is above the line
- If you are unsure
  - substitute the coordinates from a point on one side of the line into the inequality
  - determine whether or not the inequality holds true on that side
- For vertical lines:
  - the wanted region for $x less than k$ is to the left of $x equals k$
  - the wanted region for $x greater than k$ is to the right of $x equals k$
To do the shading:
- Shade the unwanted sides of each line (unless the question says otherwise)
  - You are shading away any parts you don’t want
  - This will leave behind a clear region that is the wanted region (rather than trying to look for the wanted region under multiple shades)
  - Label the wanted region R (unless the question says otherwise)
(Be careful if using graphing software, as some shade the wanted sides)

Worked Example

Show, graphically, the region that is satisfied by all three inequalities below:

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Maths Gcse Wjec-Eduqas Higher

Representing-Inequalities-As-Regions Wjec-Eduqas Higher

Finding regions using inequalities

What are 2D inequalities?

How do we draw inequalities on a graph?

Worked Example

Responses