The Cartesian plane: graphing linear relations & inequalities

A linear relation like y = 2x + 1 can be graphed on the Cartesian plane by plotting points that satisfy the equation and joining them with a straight line. The gradient (slope) tells you how steep the line is and whether it rises or falls, and the y-intercept tells you where the line crosses the y-axis. An inequality like x > 3 or y ≤ 2x − 1 describes a whole region of the plane rather than a single line — you can solve and graph it using similar techniques to an equation, but the solution is a range of values, often shown by shading a region.

Key terms

Gradient:

How steep a line is — the amount y changes for each unit x changes.

y-intercept:

The point where a line crosses the y-axis (where x = 0).

Questions

1. 1. The gradient of a line tells you:

   • How steep the line is
   • Only its colour
   • Only its length
   • Its exact position on the page

2. 2. The y-intercept is where a line crosses the:

   • y-axis
   • x-axis
   • Origin only
   • Nowhere in particular

3. 3. For y = 2x + 1, the y-intercept is:

   • 1
   • 2
   • 0
   • 3

4. 4. An inequality like x > 3 describes:

   • A whole region of values, not just one
   • Exactly one single point
   • Nothing that can be graphed
   • Only negative numbers

5. 5. For y = 2x + 1, when x = 1, y equals:

   • 3
   • 2
   • 1
   • 4

6. 6. A steeper line has a:

   • Larger gradient
   • Smaller gradient always
   • Gradient of exactly 0
   • No gradient at all

7. 7. Solutions to an inequality are often shown on a graph by:

   • Shading a region
   • A single dot only
   • Nothing visual at all
   • A different colour for the axes

8. 8. For y = 3x − 2, find y when x = 2:

   • 4
   • 3
   • 2
   • 6

9. 9. What is the y-intercept of y = −x + 5?

   • 5
   • −1
   • 0
   • −5

10. 10. A line with gradient −2 will:

    • Fall as x increases
    • Rise as x increases
    • Stay perfectly flat
    • Be impossible to graph

11. 11. Solve the inequality x + 2 > 5:

    • x > 3
    • x > 7
    • x < 3
    • x < 7

12. 12. For y = 4x, what is y when x = 0?

    • 0
    • 4
    • 1
    • Undefined

13. 13. Two lines with the same gradient but different y-intercepts are:

    • Parallel to each other
    • Always the same exact line
    • Always perpendicular
    • Impossible to graph together

14. 14. Solve the inequality 2x ≤ 10:

    • x ≤ 5
    • x ≤ 20
    • x ≥ 5
    • x ≤ 12

15. 15. Why might you check a solution to a linear equation by substituting it back into the original equation?

    • To verify the value makes both sides of the equation equal, confirming it's correct
    • Substitution never confirms whether a solution is correct
    • Checking is unnecessary once you've solved an equation
    • Substitution only works for inequalities, not equations

16. 16. Solve and graph mentally: for which values of x is 3x − 1 > 8?

    • x > 3
    • x > 9
    • x < 3
    • x > 2.33

17. 17. Why does dividing or multiplying an inequality by a negative number flip the inequality sign?

    • It reverses the order of the number line, so the relationship between the values flips
    • This is an arbitrary rule with no mathematical reason
    • It only applies to equations, not inequalities
    • The sign never actually needs to flip

18. 18. Two linear relations, y = 2x + 1 and y = 2x − 3, plotted on the same axes will:

    • Never intersect, since they are parallel with the same gradient
    • Always intersect at the origin
    • Be the exact same line
    • Intersect at exactly one random point

19. 19. A mobile phone plan costs $20 plus $0.50 per GB used, modelled as y = 0.5x + 20. What does the gradient (0.5) represent in context?

    • The cost per additional GB of data used
    • The fixed monthly fee
    • The total number of GB included
    • The maximum possible cost

20. 20. Why might a company use a linear graph to model costs against production volume?

    • It visually shows the relationship and lets you predict costs at different volumes
    • Linear graphs cannot represent real-world costs
    • Graphing costs has no practical business use
    • Only non-linear relationships can be graphed meaningfully

21. 21. Understanding how to graph linear relations and inequalities mainly helps you to:

    • Visualise and interpret relationships between two variables, including ranges of possible values
    • Avoid ever representing maths visually
    • Assume every relationship between variables is identical
    • Ignore the meaning of a gradient or intercept