The difference between the areas of the figures is less than 4.

An absolute value inequality that represents this situation is ___

The solution of the inequality is ___

Answer:

1. |40 – 4x| < 4

2. 9 < x < 11

Step-by-step explanation:

We'll begin by calculating the area of each figure. This can be obtained as follow:

For triangle:

Area = ½ × base × height

Base = 8

Height = 6

Area =?

Area = ½ × 8 × 6

Area = 4 × 6

Area = 24

For Rectangle:

Area = length × breadth

Length = x – 4

Breadth = 4

Area =?

Area = (x – 4) × 4

Area = 4x – 16

Summary:

Area of triangle (Aₜ) = 24

Area of rectangle (Aᵣ) = 4x – 16

1. Writing an absolute value for the inequality.

From the question given above, we were told that the difference between the areas of the figures is less than 4. This can be written as:

Aₜ – Aᵣ < 4

Area of triangle (Aₜ) = 24

Area of rectangle (Aᵣ) = 4x – 16

24 – (4x – 16) < 4

Simplify

24 – (4x – 16) < 4

Clear bracket

24 – 4x + 16 < 4

24 + 16 – 4x < 4

40 – 4x < 4

Thus, we can write the absolute value for the above inequality as:

|40 – 4x| < 4

2. Determination of the solution of the inequality.

This is illustrated below:

|40 – 4x| < 4

Thus the above equation can be written as

– 4 < 40 – 4x < 4

Subtract 40 from both side

– 4 – 40 < 40 – 4x – 40 < 4 –40

– 44 < – 4x < – 36

Divide through by –4

– 44/–4 < x < – 36/–4

11 > x > 9

Note: the inequality sign changed because we divided by a negative number

Thus, 11 > x > 9 becomes 9 < x < 11

Therefore, the solution to the inequality is 9 < x < 11

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