Introduction to Inequalities

Show your work for each problem.

1.
Which symbol means 'greater than'?
1. a) $>$
2. b) $=$
3. c) $\neq$
4. d) $<$
Answer: $>$
The symbol $>$ means 'greater than'. The open side of the symbol faces the larger value.
2.
Is the statement $5 < 9$ true or false?
1. a)True
2. b)Cannot be determined
3. c)False
4. d)Neither
Answer: True
True. The statement $5 < 9$ reads '5 is less than 9', which is correct because 5 is smaller than 9.
3.
Fill in $<$ or $>$ : $12 \,\square\, 7$
Answer: >
$12 > 7$ because 12 is greater than 7. The symbol opens toward the larger number.
4.
Fill in $<$ or $>$ : $3 \,\square\, 8$
Answer: <
$3 < 8$ because 3 is less than 8. The pointed end of the symbol points to the smaller number.
5.
What does the symbol $\leq$ mean?
1. a)Not equal to
2. b)Less than
3. c)Greater than or equal to
4. d)Less than or equal to
Answer: Less than or equal to
$\leq$ means 'less than or equal to'. The line underneath the $<$ adds the possibility of equality.
6.
Which statement is TRUE?
1. a) $-5 \geq -2$
2. b) $-5 < -2$
3. c) $-5 = -2$
4. d) $-5 > -2$
Answer: $-5 < -2$
$-5 < -2$ is true. On the number line, $-5$ is to the LEFT of $-2$ , so $-5$ is smaller. Think: $-5°C$ is colder than $-2°C$ .
7.
Fill in $<$ or $>$ : $-8 \,\square\, -3$
Answer: <
$-8 < -3$ because $-8$ is further left on the number line. Think of temperature: $-8°C$ is colder (smaller) than $-3°C$ .
8.
Is the statement $7 \leq 7$ true or false?
1. a)False
2. b)Only true if less than
3. c)True
4. d)Only true if equal
Answer: True
True! $7 \leq 7$ is true because 7 equals 7. The $\leq$ symbol means 'less than OR equal to', and the 'equal to' part is satisfied.
9.
A ride requires you to be at least 130 cm tall. Which inequality describes the allowed height $h$ (in cm)?
1. a) $h < 130$
2. b) $h \geq 130$
3. c) $h \leq 130$
4. d) $h > 130$
Answer: $h \geq 130$
$h \geq 130$ : the height must be 130 cm or more. 'At least' becomes $\geq$ because exactly 130 cm is allowed; $h > 130$ would wrongly turn away someone who is exactly 130 cm tall.
10.
Order from least to greatest: $-4$ , $1$ , $-7$ , $3$
Answer: -7, -4, 1, 3
1. Which number is furthest left on the number line (smallest)? -7
2. Which is the next smallest number? -4
3. Which positive number comes next? 1
4. Which is the largest (furthest right)? 3
11.
You have 30 dollars and want to buy a book for 12 dollars and a toy. What's the maximum price for the toy? Write and solve the inequality.
Answer: 18
1. What is your total budget? 30
2. How much does the book cost? 12
3. If the toy costs $t$ dollars, write the inequality: $12 + t \leq$ ? 30
4. Solve for $t$ : What is the maximum value? 18
12.
Which value does NOT satisfy $x \geq -2$ ?
1. a) $x = 0$
2. b) $x = 10$
3. c) $x = -5$
4. d) $x = -2$
Answer: $x = -5$
$x = -5$ does NOT satisfy $x \geq -2$ because $-5 < -2$ . The other values (0, -2, 10) are all greater than or equal to -2.
13.
A pool gives free entry to children under 10 years old. Which inequality describes the ages $a$ that get free entry?
1. a) $a > 10$
2. b) $a \geq 10$
3. c) $a < 10$
4. d) $a \leq 10$
Answer: $a < 10$
$a < 10$ : a 9-year-old gets in free, but a 10-year-old does NOT, so 10 itself is excluded and we use $<$ , not $\leq$ .
14.
A speed limit sign shows 'Maximum 80 km/h'. A car is traveling at 85 km/h. Is the driver breaking the law? Explain using inequalities.
Answer: Yes, because $85 > 80$
1. What does 'maximum 80 km/h' mean? Write the rule as an inequality using $v$ for speed. v ≤ 80
2. What speed is the car traveling? 85
3. Is $85 \leq 80$ true or false? false
4. Is the driver breaking the law? yes
15.
Put these numbers in order from greatest to least: $2$ , $-6$ , $0$ , $-1$ , $5$
Answer: 5, 2, 0, -1, -6
1. Which number is the greatest (furthest right on number line)? 5
2. Which is the next greatest? 2
3. What number comes between positive and negative? 0
4. Which negative number is closer to zero? -1
5. Which is the smallest (furthest left)? -6
16.
Which inequality correctly represents: 'You must be at least 13 but under 18 to join the youth club'? Use $a$ for age.
1. a) $13 < a < 18$
2. b) $13 < a \leq 18$
3. c) $13 \leq a < 18$
4. d) $13 \leq a \leq 18$
Answer: $13 \leq a < 18$
$13 \leq a < 18$ is correct. 'At least 13' means age 13 is allowed ( $\geq$ ), and 'under 18' means 18 is NOT allowed ( $<$ ).