Question

Generalize What is the difference between an
equation with a variable and an inequality with
a variable?
3. Would it be more efficient to use an inequality
or to list all of the quantities less than 6? Explain.
4. Generalize How are the symbols for greater
than ( 7 ) and greater than or equal to ( Ú )
related?
Do You Know How?
In 5–12, write an inequality for each situation.
5. A number, n, is greater than 22.
6. The value, v, does not equal 2 1
2 .
7. Sally’s age, a, is at most 15.
8. The width of the picture, w, is shorter than
8.5 inches.
9. Steve’s height, h, is at least 48 inches.
10. Vera’s baby brother’s age, b, is not 24 months.
11. The number of quarters, q, in the jar is less
than 75.
12. The length of the fish a fisherman catches, f,
must be at least 10 inches for him to keep it.

Answer 1

1. Generalize — difference between an equation with a variable and an inequality with a variable
- An equation (with =) states two expressions are exactly equal; its solutions are the value(s) of the variable that make the two sides equal (often a single value or a finite set).
- An inequality (using <, >, ≤, ≥, ≠) describes a range of possible values for the variable; its solutions are typically an interval or set of many values (shown on a number line or as an interval).

3. Would it be more efficient to use an inequality or to list all of the quantities less than 6? Explain.
- Use an inequality. For real numbers the set “numbers less than 6” is infinite, so an inequality x < 6 efficiently represents them all. (If you were restricted to a small finite domain, e.g. whole numbers less than 6, listing {0,1,2,3,4,5} is possible, but in general the inequality is more concise.)

4. Generalize — how are the symbols for greater than (>) and greater than or equal to (≥) related?
- “≥” means “> or =.” So a ≥ b is true whenever a > b or a = b. On a number line, ≥ includes the point of equality; > does not.

Do You Know How? Write an inequality for each situation (5–12)
5. n > 22
6. v ≠ 2 1/2 (v is not equal to 2 1/2)
7. a ≤ 15 (Sally’s age is at most 15)
8. w < 8.5 (width is shorter than 8.5 inches)
9. h ≥ 48 (Steve’s height is at least 48 inches)
10. b ≠ 24 (baby brother’s age is not 24 months)
11. q < 75 (number of quarters is less than 75)
12. f ≥ 10 (fish must be at least 10 inches to keep)