Describe each of the following sets - Junior Cycle Mathematics - Question 7 - 2019

The set of integers, ( \mathbb{Z} ), includes all positive and negative whole numbers, as well as zero. Therefore, it can be described as:

[ \mathbb{Z} = { \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots } ]

To graph the inequality (-3 < x \leq 2) on the number line, we represent the values of (x) that satisfy this condition:

• The value (-3) is represented with an open circle (not included).
• The value (2) is represented with a closed circle (included).

Thus, the number line would look as follows:

  -4   -3   -2   -1    0    1    2    3    4
       o==============)

This indicates that (x) can take any value between (-3) (not included) to (2) (included).

To solve this compound inequality, we will tackle each part separately.

1. First part: (-7 < 8 - 3g)

   • Subtract 8 from both sides: [-7 - 8 < -3g \Rightarrow -15 < -3g]
   • Divide both sides by -3 (remember to reverse the inequality): [5 > g \quad or \quad g < 5]

2. Second part: (8 - 3g \leq 11)

   • Subtract 8 from both sides: [-3g \leq 3]
   • Divide by -3 (reverse the inequality): [g \geq -1]

Combining the two results: [-1 \leq g < 5]

This means that the solution set for (g) is given by: [g \in [-1, 5)]