Algebra Know-It-ALL

622 Worked-Out Solutions to Exercises: Chapters 11 to 19

9. We must simplify the inequality so that y appears all by itself on the left side of the
   “not equal” symbol, and a plain numeral appears all by itself on the right. Here’s the
   inequality again, for reference:

y/2≠ 4 y+ 7

First, let’s multiply through by 2. That gives us

(y/2)× 2 ≠ (4y+ 7) × 2

which multiplies out to

y≠ 8 y+ 14

Now, let’s subtract 8y from each side. That gives us

y− 8 y≠ 8 y+ 14 − 8 y

which simplifies to

− 7 y≠ 14

We can divide this through by −7 to get

(− 7 y)/(−7)≠ 14 /(−7)

which simplifies to

y≠− 2

The original inequality holds true for all values of y except −2.

10. We must simplify the inequality so that z appears all by itself on the left side of the
    “smaller than or equal” symbol, and a plain numeral appears all by itself on the right.
    Here’s the inequality again, for reference:

z/(−3)≤ 6 z+ 6

Let’s multiply through by −3, remembering that we must reverse the sense of the inequal-

ity whenever we multiply through by a negative. That gives us

[z/(−3)]× (−3)≥ (6z+ 6) × (−3)

which simplifies to

z≥− 18 z− 18