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Solve Inequalities with Two Variables

In this worksheet, students will practise drawing inequalities on a set of axes.

'Solve Inequalities with Two Variables' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Inequalities

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

At GCSE you have to be able to illustrate inequalities. When you have a single variable inequality (such as 2x + 1 > 7) you can do this with a number line. However, what do we do when we have inequalities that contain two variables.

We use a set of axes.

The difference between greater than and greater than or equal to.

When we draw inequalities on a number line, we use a shaded circle for greater than or equal to and an unshaded on for greater than.

When we are drawing on a set of axes, we have to change this slightly.

Greater Than/Less than – We use a dashed line

Greater Than or Equal to/Less than or equal to – We use a solid line.

 

Example 1: Draw the inequality y > 3x + 1

The first step here is to work out if we need a solid or a dashed line. As the inequality is a greater then, we use a dashed line.

The next step is to work out the line. To do this, we ignore the inequality and draw the line y = 3x + 1

Once we have drawn the line, we need to decide which side to shade to illustrate the inequality. To do this, we pick a point (I think 0,0 is the best one), plug this into the inequality and see if the point satisfies the inequality.

We know at (0,0), x = 0 and y = 0

Y > 3x + 1

0 > 3(0) + 1

0 > 1

We can see very clearly that this isn’t true, so we shade above the line (if it was true, we would shade below)

 

Example 2: Draw the inequality 5x + 2y ≤ 10

Once again, we pick our line type first. As this is less than or equal to, we use a solid line.

Once again, we need to test to see which side we need to shade. I’m going to use (0,0) again.

5x + 2y ≤ 10

5(0)+ 2(0) ≤ 10

0 ≤ 10

Because this is true, we know that this is the side of the line we need to shade.

 

Example 3: Draw the inequality -2 < x ≤ 1

It is really common with this style of questions that you have to draw single variable inequalities on a set of axes. To do this, we have to look at the inequality as two lines. -2 < x and x ≤ 1.

If we draw this on a set of axes, we get…

 

We don’t need to test this one as the inequality is quite clear. We are looking for the points between -2 and 1 so our shading would look like…

To illustrate an inequality on a set of axes...

Complete the following sentence...

Which of the following graphs is correct for y = 2x + 1?



Which of the following graphs is correct for y < 2x + 1?

 



Match the inequality with the function.

Column A

Column B

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y> 3x + 1

');" onmouseout="tooltip.hide();">

y < 2x – 3

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y > x + 4

Match the inequality with the function.

 

Column A

Column B

y > x

');" onmouseout="tooltip.hide();">

y < x

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y ≥ x

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y ≤ x

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Which of these coordinates satisfy the inequality y > 4x – 5?

(0,0)

(2,3)

(0,4)

(1,-1)

Which of these coordinates satisfy the inequality 2x + 3y ≤ 6

(0,0)

(2,3)

(0,4)

(1,-1)

Which of these graphs correctly illustrates -4 < x ≤ 5

 



Which of these graphs correctly illustrates 3x + 2y < 6

 



  • Question 1

To illustrate an inequality on a set of axes...

CORRECT ANSWER
EDDIE SAYS
All inequalities we have to draw at GCSE are linear, so we first draw a linear graph.
  • Question 2

Complete the following sentence...

CORRECT ANSWER
EDDIE SAYS
If an inequality is greater than or equal to, we need to say that solutions can lie on the line. This can only happen if the line is solid. If the line is dashed, solutions cannot be on the line.
  • Question 3

Which of the following graphs is correct for y = 2x + 1?

CORRECT ANSWER

EDDIE SAYS
Its really important in these questions that you read everything closely. The question doesn’t ask you to draw an inequality, it asks you to draw a linear equation. For these, you would always draw with a solid line.
  • Question 4

Which of the following graphs is correct for y < 2x + 1?

 

CORRECT ANSWER

EDDIE SAYS
To work out which side we need to shade, we must test a point. If we use (0,0) x = 0 and y = 0 Substituting these into the inequality.. Y < 2x + 1 0 < 2(0) + 1 0 < 1 Because this is true, the point (0,0) is on the side of the line that satisfies the inequality so we shade
  • Question 5

Match the inequality with the function.

CORRECT ANSWER

Column A

Column B

');" onmouseout="tooltip.hide();">

y> 3x + 1

');" onmouseout="tooltip.hide();">

y < 2x – 3

');" onmouseout="tooltip.hide();">

y > x + 4
EDDIE SAYS
Do we even need to think about the inequalities here? Let’s remember that in the form y = mx + c, the value of C is the y intercept so we can just match them up.
  • Question 6

Match the inequality with the function.

 

CORRECT ANSWER

Column A

Column B

y > x

');" onmouseout="tooltip.hide();">

y < x

');" onmouseout="tooltip.hide();">

y ≥ x

');" onmouseout="tooltip.hide();">

y ≤ x

');" onmouseout="tooltip.hide();">

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EDDIE SAYS
This question is all about the shading and the lines. Remember that the lines have different meanings. A solid line is one that is ‘…or equal to’ so we can deal with those first. If you test a point, you will find that greater than is above the line. You should be able to work out the rest from there
  • Question 7

Which of these coordinates satisfy the inequality y > 4x – 5?

CORRECT ANSWER
(0,0)
(0,4)
EDDIE SAYS
For each of these, we just need to bang the numbers in. For example (2,3) Y > 4x – 5 3 > 4(2) – 5 3 > 3 This doesn’t work because 3 isn’t greater than 3.
  • Question 8

Which of these coordinates satisfy the inequality 2x + 3y ≤ 6

CORRECT ANSWER
(0,0)
(1,-1)
EDDIE SAYS
This one is another where you plug in the numbers. If your answer is true, the coordinates satisfy the inequalitiy. i.e. (2,3) 2(2) + 3(3) ≤ 6 4 + 9 ≤ 6 13≤6 As this is not true, that point does not satisfy the inequality.
  • Question 9

Which of these graphs correctly illustrates -4 < x ≤ 5

 

CORRECT ANSWER

EDDIE SAYS
The most common mistake here is to get the lines the wrong way round. If the inequality involves x, the lines are vertical. If they involve y, they will be horizontal.
  • Question 10

Which of these graphs correctly illustrates 3x + 2y < 6

 

CORRECT ANSWER

EDDIE SAYS
Again, the easiest way to test this is to test a point. If I use (0,0) 3x + 2y < 6 3(0) + 2(0) < 6 0 < 6 We now know that (0,0) is on the shaded side so the correct one must be…
---- OR ----

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