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

The set of integers, denoted as ℤ, includes all whole numbers, both positive and negative, as well as zero:

ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}

This means integers encompass positive numbers, negative numbers, and zero, excluding fractions or decimals.

To graph the inequality −3 < x ≤ 2 on a number line:

1. Draw a horizontal line and mark the points −3 and 2.
2. Use an open circle at −3 to indicate it is not included in the solution (since x is greater than −3).
3. Use a closed circle at 2 to indicate it is included in the solution (since x is less than or equal to 2).
4. Shade the region between −3 and 2, including 2.

To solve the compound inequality −7 < 8 − 3g ≤ 11, follow these steps:

1. Separate the compound inequality into two parts:

   • First part:

     $-7 < 8 - 3g$

   • Second part:

     $8 - 3g ≤ 11$

2. Solve the first part:

   • Subtract 8 from both sides:

     $-7 - 8 < -3g$

     simplifies to

     $-15 < -3g$

   • Divide by -3 (remember to flip the inequality):

     $g < 5$

3. Solve the second part:

   • Subtract 8 from both sides:

     $8 - 3g - 8 ≤ 11 - 8$

     simplifies to

     $-3g ≤ 3$

   • Divide by -3 (again, flip the inequality):

     $g ≥ -1$

4. Combine the results:

   The final solution is:

   $-1 ≤ g < 5$