Inequalities Worked Examples for IGCSE Maths

Question

Solve 3x - 7 > 2x + 5.

Solution

1. 1

   Collect x terms on one side

   3x - 2x > 5 + 7

   Subtract 2x from both sides and add 7 to both sides.

2. 2

   Simplify

   x > 12

   Combine like terms.

Question

Solve -3 < 2x + 1 <= 9 and list the integer values.

Solution

1. 1

   Subtract 1 from all parts

   -3 - 1 < 2x <= 9 - 1, so -4 < 2x <= 8

   Apply the same operation to all three parts.

2. 2

   Divide all parts by 2

   -2 < x <= 4

   Divide each part by 2.

3. 3

   List integer values

   x can be -1, 0, 1, 2, 3, 4

   -2 is not included (strict inequality) but 4 is included (<=).

Question

On a grid, show the region satisfying y >= 1, x + y <= 6, and y <= 2x.

Solution

1. 1

   Draw y = 1

   Horizontal line through y = 1. Shade ABOVE (y >= 1). Solid line.

   y >= 1 means all points on or above the line y = 1.

2. 2

   Draw x + y = 6

   Line through (0,6) and (6,0). Shade BELOW/LEFT (x+y <= 6). Solid line.

   Test (0,0): 0+0 = 0 <= 6, true, so shade the side containing (0,0).

3. 3

   Draw y = 2x

   Line through (0,0) and (1,2). Shade BELOW/RIGHT (y <= 2x). Solid line.

   Test (3,1): 1 <= 6, true, so shade the side containing (3,1).

4. 4

   Identify the region

   The required region is the triangle where all three conditions are satisfied simultaneously.

   The feasible region is the overlap of all three shaded regions.

Question

Solve x squared - 5x + 6 <= 0.

Solution

1. 1

   Factorise

   x squared - 5x + 6 = (x-2)(x-3) = 0 when x = 2 or x = 3

   First find where the expression equals zero.

2. 2

   Determine the solution region

   The parabola y = x squared - 5x + 6 opens upwards (positive coefficient of x squared). It is negative (below the x-axis) between the roots.

   For <= 0 with an upward parabola, the solution is between the roots.

3. 3

   Write the solution

   2 <= x <= 3

   Include the endpoints because the inequality is <=.