Compound Inequalities

Show your work for each problem.

1.
Which compound inequality represents " is greater than AND less than "?
1. a)
2. b) OR
3. c)
4. d) AND
Answer:
" AND " can be written compactly as . This represents all values between and .
2.
What is the interval notation for OR ?
1. a)
2. b)
3. c)
4. d)
Answer:
OR requires union (). Since and include the endpoints, we use square brackets: .
3.
Solve: . What is the upper bound of the solution?
Answer: 3
Subtract 2: , giving . The upper bound is .
4.
Does AND have a solution? Answer 'yes' or 'no'.
Answer: no
No. For AND, both conditions must be true simultaneously. No number can be both greater than 4 and less than 1. The solution set is empty ().
5.
Solve OR . Which is the correct solution?
1. a) OR
2. b) OR
3. c)
4. d) AND
Answer: OR
First: . Second: . Combined: OR .
6.
Solve:
Answer: -2 <= x <= 3
1. Subtract 2 from all three parts. What is the left side? -6
2. What is the right side after subtracting 2? 9
3. Now divide all parts by 3. What is the lower bound for ? -2
4. What is the upper bound for ? 3
7.
Solve: OR . What is the boundary value for the first inequality?
Answer: -2
. The boundary value is .
8.
Solve: AND
Answer: -3 <= x <= 4
1. Write as a combined inequality: . Subtract 5 from all parts. What is the left side? -8
2. What is the right side after subtracting 5? 6
3. Now we have . Divide by and FLIP the signs. What is the new left bound? -3
4. What is the new right bound? 4
9.
A theme park ride requires height to be at least 120 cm but no more than 200 cm. Which inequality represents this?
1. a)
2. b) OR
3. c) AND
4. d)
Answer:
'At least 120' means . 'No more than 200' means . Combined with AND: .
10.
Write the interval notation for . Use parentheses ( ), brackets [ ], and a comma.
Answer: (1, 6]
means 1 is not included (use parenthesis). means 6 is included (use bracket). Answer: .
11.
Solve: AND
Answer: 1 < x < 7
1. Solve the first inequality. Multiply both sides by 2: 6
2. Add 1 to both sides. What is less than? 7
3. Now solve the second. Multiply both sides by 3: 3
4. Subtract 2. What is greater than? 1
12.
What is the solution to OR ?
1. a)All real numbers ()
2. b) OR
3. c)
4. d)No solution
Answer: All real numbers ()
Every real number is either or (or both). For example: (true), (true), satisfies both. The solution is all real numbers.
13.
Solve:
Answer: -2 < x < 8/3
1. Multiply all three parts by 2. What is the left side? -4
2. What is the right side after multiplying by 2? 10
3. Now we have . Subtract 4 from all parts. What is the new left side? -8
4. What is the new right side? 6
5. We have . Divide by and flip signs. What is the lower bound for ? -2
14.
A product passes quality control if its weight is between 95 and 105 grams (inclusive). Write the compound inequality for passing weight . Use the format: a <= w <= b
Answer: 95 <= w <= 105
Weight must be at least 95 () AND at most 105 (). Combined: .
15.
Solve: OR
Answer: x < 3 OR x > 3
1. Expand the first inequality: . Add 2. What is less than? 6
2. Divide by 2. What is less than? 3
3. Now expand the second: . Subtract 6. What is greater than? 9
4. Divide by 3. What is greater than? 3
16.
Graph the solution to on a number line. Describe the endpoints.
Answer: open at -2, closed at 4
1. Look at the left endpoint . The symbol is (strictly less than). Should be included? no
2. For points NOT included, we use an open or closed circle? open
3. Look at the right endpoint . The symbol is . Should be included? yes
4. For points that ARE included, we use an open or closed circle? closed