Exercise 12 Page 204

We have the following compound inequality. 5 ≤ y+2 ≤ 11 Let's split it into two separate inequalities. 5 ≤ y+2 and y+2 ≤ 11Now we can solve them separately and later compare their solution sets.

First Inequality

We can solve inequalities just like ordinary equations. By subtracting 2 from both sides of the first inequality, we can eliminate 2 on the right-hand side which will isolate y.

The first inequality is satisfied by all values greater than or equal to 3. Note that y can equal 3 as the inequality is non-strict.

Second Inequality

Now we can solve the second inequality. Again, by subtracting 2 from both sides, we can eliminate 2 on the left-hand side and isolate y.

The second inequality is true for numbers less than or equal to 9. Notice that y can equal 9 as the inequality is non-strict.

Combining Solution Sets

Finally, we can combine the obtained solution sets. The first inequality describes all values to the right of 3, including 3.

The second inequality describes all values to the left of 9, including 9.

The intersection of these sets, 3 ≤ y ≤ 9, describes the solutions of the compound inequality.