3. make each equation true by filling the black boxes with +, −, ×, or ÷ and adding brackets () where necessary.
a. $-2 \boldsymbol{\square} 5 \boldsymbol{\square} -8 = 11$
b. $-4 \boldsymbol{\square} 7 \boldsymbol{\square} 9 = 2$
c. $3 \boldsymbol{\square} -2 \boldsymbol{\square} -5 = -1$
4. write an equation like in question 3. trade your equation with a classmate. make your partner’s equation true.
5. the table shows the size of the deer population in a national park from 2016 to 2021. determine the average (mean) change per year in the size of the deer population.
| year | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 |
|------------|------|------|------|------|------|------|
| number of deer | 53 | 47 | 44 | 51 | 50 | 43 |
what does the sign of the mean change per year tell you about the deer population overall?
6. describe how to determine the next number in each pattern. then continue the pattern for 2 more terms.
a) $-14, -11, -8, -5, \dots$
b) $9, 4, -1, -6, \dots$
c) $2, -4, 8, -16, \dots$
7. evaluate each expression for the given values.
a) $3m - 12$ for $m = -9$
b) $-7a + 3(a - 8)$ for $a = 5$
c) $-2(6 + 7b)^2$ for $b = -2$
d) $4(w - 5)(6 + 3u)$ for $w = -3$ and $u = -4$
8. identify the error(s) in each solution. then write the correct solution.
a) $5 - 124 - (-7)$
$= -7(4 + 7)$
$= -7(11)$
$= -77$
b) $(-6)^2 - (-4)$
$= -36 - 4$
$= -40$

Question

3. make each equation true by filling the black boxes with +, −, ×, or ÷ and adding brackets ()  where necessary.
   a. $-2 \boldsymbol{\square} 5 \boldsymbol{\square} -8 = 11$
   b. $-4 \boldsymbol{\square} 7 \boldsymbol{\square} 9 = 2$
   c. $3 \boldsymbol{\square} -2 \boldsymbol{\square} -5 = -1$
4. write an equation like in question 3. trade your equation with a classmate. make your partner’s equation true.
5. the table shows the size of the deer population in a national park from 2016 to 2021. determine the average (mean) change per year in the size of the deer population.
   | year       | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 |
   |------------|------|------|------|------|------|------|
   | number of deer | 53   | 47   | 44   | 51   | 50   | 43   |
   what does the sign of the mean change per year tell you about the deer population overall?
6. describe how to determine the next number in each pattern. then continue the pattern for 2 more terms.
   a) $-14, -11, -8, -5, \dots$
   b) $9, 4, -1, -6, \dots$
   c) $2, -4, 8, -16, \dots$
7. evaluate each expression for the given values.
   a) $3m - 12$ for $m = -9$
   b) $-7a + 3(a - 8)$ for $a = 5$
   c) $-2(6 + 7b)^2$ for $b = -2$
   d) $4(w - 5)(6 + 3u)$ for $w = -3$ and $u = -4$
8. identify the error(s) in each solution. then write the correct solution.
   a) $5 - 124 - (-7)$
      $= -7(4 + 7)$
      $= -7(11)$
      $= -77$
   b) $(-6)^2 - (-4)$
      $= -36 - 4$
      $= -40$

Accepted Answer

### Problem 3a:
# Explanation:
## Step1: Analyze operations and brackets
We need to find operations (+, -, ×, ÷) and brackets to make $-2 \square 5 \square -8 = 11$ true. Let's try $-2 - 5 \times (-8)$.
## Step2: Apply order of operations (PEMDAS)
First, multiply: $5 \times (-8)= -40$, then subtract: $-2 - (-40)= -2 + 40 = 38$ (not 11). Try $-2 + 5 - (-8)$.
## Step3: Simplify
$-2 + 5 = 3$, $3 - (-8)= 3 + 8 = 11$. Wait, no brackets needed? Wait, $-2 + 5 - (-8)= -2 + 5 + 8 = 11$. Wait, maybe $-2 - (5 \times (-8))$ was wrong. Wait, let's check again. Wait, $-2 - 5 \times (-8)= -2 + 40 = 38$. Wait, maybe $-2 + (5 - (-8))$? $5 - (-8)=13$, $-2 +13=11$. Yes! So $-2 + (5 - (-8)) = 11$. Wait, but the original is $-2 \square 5 \square -8$. So $-2 + (5 - (-8))$ can be written as $-2 + 5 - (-8)$ (since subtracting a negative is adding positive). Wait, maybe the correct is $-2 - 5 \times (-8)$ was miscalculated? No, $-2 - (5 \times (-8))= -2 + 40 = 38$. Wait, let's try $-2 \times 5 - (-8)$? $-10 +8=-2$. No. Wait, $-2 - 5 + (-8)$? $-15$. No. Wait, $-2 + 5 \times (-8)$? $-2 -40=-42$. No. Wait, maybe $-2 - (5 + (-8))$? $5 + (-8)=-3$, $-2 - (-3)=1$. No. Wait, the correct one is $-2 + (5 - (-8)) = -2 + 13 = 11$, so the operations are $+$ and $-$ with a bracket: $-2 + (5 - (-8)) = 11$, or without brackets: $-2 + 5 - (-8) = 11$ (since subtracting a negative is adding, so $-2 + 5 + 8 = 11$).

### Problem 3b:
# Explanation:
## Step1: Analyze operations
We need $-4 \square 7 \square 9 = 2$. Let's try $(-4 + 7) - 9$? $3 -9=-6$. No. $(-4 + 7) + 9=12$. No. $-4 + (7 - 9)= -4 -2=-6$. No. $-4 \times 7 + 9= -28 +9=-19$. No. $-4 + 7 \times 9= -4 +63=59$. No. Wait, $(-4) + (7 - 9)$ no. Wait, $(-4) \times (7 - 9)$? $7 -9=-2$, $-4 \times (-2)=8$. No. Wait, $(-4) + (7 + 9)=12$. No. Wait, $(-4) - (7 - 9)= -4 - (-2)= -2$. No. Wait, $(-4) \div (7 - 9)$? $7 -9=-2$, $-4 \div (-2)=2$. Yes! So $-4 \div (7 - 9) = 2$.

### Problem 3c:
# Explanation:
## Step1: Analyze operations
We need $3 \square -2 \square -5 = -1$. Let's try $3 + (-2) + (-5)=3 -2 -5=-4$. No. $3 - (-2) + (-5)=3 +2 -5=0$. No. $3 + (-2) - (-5)=3 -2 +5=6$. No. $3 - (-2) - (-5)=3 +2 +5=10$. No. $3 \times (-2) + (-5)= -6 -5=-11$. No. $3 + (-2) \times (-5)=3 +10=13$. No. $3 - (-2) \times (-5)=3 -10=-7$. No. Wait, $3 + (-2) + (-5)$ no. Wait, $3 - 2 - 5$? $3 -2=1$, $1 -5=-4$. No. Wait, $3 + (-2) - 5=3 -2 -5=-4$. No. Wait, $3 - (-2) + 5=10$. No. Wait, $3 + (-2) \times (-5)$ no. Wait, maybe $3 + (-2) + (-5)$ is wrong. Wait, $3 - 2 + (-5)=3 -2 -5=-4$. No. Wait, $3 + (-2) - (-5)=3 -2 +5=6$. No. Wait, $3 - (-2) - 5=3 +2 -5=0$. No. Wait, $3 \times (-2) - (-5)= -6 +5=-1$. Yes! So $3 \times (-2) - (-5) = -6 +5 = -1$.

### Problem 5:
# Explanation:
## Step1: Calculate total change
First, find the change in population each year:
- 2016 to 2017: $47 - 53 = -6$
- 2017 to 2018: $44 - 47 = -3$
- 2018 to 2019: $51 - 44 = 7$
- 2019 to 2020: $50 - 51 = -1$
- 2020 to 2021: $43 - 50 = -7$
## Step2: Sum the changes
Total change = $-6 + (-3) + 7 + (-1) + (-7) = -6 -3 +7 -1 -7 = (-6 -3 -1 -7) +7 = (-17) +7 = -10$
## Step3: Calculate number of years
From 2016 to 2021, there are $2021 - 2016 = 5$ years of change (since we have 6 data points, the number of intervals is 5).
## Step4: Calculate mean change
Mean change = $\frac{\text{Total change}}{\text{Number of years}} = \frac{-10}{5} = -2$
## Step5: Interpret the sign
A negative mean change per year means the deer population is decreasing overall.

### Problem 6a:
# Explanation:
## Step1: Identify the pattern
The sequence is $-14, -11, -8, -5, \dots$. The difference between consecutive terms: $-11 - (-14) = 3$, $-8 - (-11) = 3$, $-5 - (-8) = 3$. So it's an arithmetic sequence with common difference $+3$.
## Step2: Find next terms
Next term after $-5$: $-5 + 3 = -2$
Term after $-2$: $-2 + 3 = 1$

### Problem 6b:
# Explanation:
## Step1: Identify the pattern
The sequence is $9, 4, -1, -6, \dots$. The difference between consecutive terms: $4 - 9 = -5$, $-1 - 4 = -5$, $-6 - (-1) = -5$. So it's an arithmetic sequence with common difference $-5$.
## Step2: Find next terms
Next term after $-6$: $-6 + (-5) = -11$
Term after $-11$: $-11 + (-5) = -16$

### Problem 6c:
# Explanation:
## Step1: Identify the pattern
The sequence is $2, -4, 8, -16, \dots$. The ratio between consecutive terms: $\frac{-4}{2} = -2$, $\frac{8}{-4} = -2$, $\frac{-16}{8} = -2$. So it's a geometric sequence with common ratio $-2$.
## Step2: Find next terms
Next term after $-16$: $-16 \times (-2) = 32$
Term after $32$: $32 \times (-2) = -64$

### Problem 7a:
# Explanation:
## Step1: Substitute $m = -9$ into $3m - 12$
$3(-9) - 12$
## Step2: Calculate
$3(-9) = -27$, so $-27 - 12 = -39$

### Problem 7b:
# Explanation:
## Step1: Simplify the expression $-7a + 3(a - 8)$
First, distribute the 3: $-7a + 3a - 24$
## Step2: Combine like terms
$(-7a + 3a) - 24 = -4a - 24$
## Step3: Substitute $a = 5$
$-4(5) - 24 = -20 - 24 = -44$

### Problem 7c:
# Explanation:
## Step1: Substitute $b = -2$ into $-2(6 + 7b)2$ (assuming it's $-2(6 + 7b) \times 2$ or $-2(6 + 7b)^2$? Wait, the original is $-2(6 + 7b)2$, maybe it's $-2 \times (6 + 7b) \times 2$ or $-2(6 + 7b)^2$. Wait, likely a typo, maybe $-2(6 + 7b)^2$? Wait, no, the user wrote $-2(6 + 7b)2$. Maybe it's $-2 \times (6 + 7b) \times 2$. Let's check. If $b = -2$, then $6 + 7(-2) = 6 -14 = -8$. Then $-2 \times (-8) \times 2 = 16 \times 2 = 32$. Wait, or maybe $-2(6 + 7b) \times 2$ is $-4(6 + 7b)$. Then $-4(6 -14)= -4(-8)=32$.

### Problem 7d:
# Explanation:
## Step1: Substitute $w = -3$ and $u = -4$ into $4(w - 5)(6 + 3u)$
First, calculate $w - 5 = -3 -5 = -8$
Calculate $6 + 3u = 6 + 3(-4) = 6 -12 = -6$
## Step2: Multiply the terms
$4 \times (-8) \times (-6) = 4 \times 48 = 192$

### Problem 8a:
# Explanation:
## Step1: Identify the error
The original solution: $5 - 12[4 - (-7)] = -7(4 + 7)$ is wrong. The first step should be to calculate the bracket first: $4 - (-7) = 11$, then multiply by 12: $12 \times 11 = 132$, then subtract from 5: $5 - 132 = -127$. The error was combining 5 and -12 incorrectly (shouldn't subtract 5 and 12 first, order of operations: brackets first, then multiplication, then subtraction).
## Step2: Correct solution
$5 - 12[4 - (-7)] = 5 - 12[11] = 5 - 132 = -127$

### Problem 8b:
# Explanation:
## Step1: Identify the error
The original solution: $(-6)^2 - (-4) = -36 - 4$ is wrong. $(-6)^2 = 36$ (since squaring a negative gives positive), and $-(-4) = +4$. So it should be $36 + 4 = 40$.
## Step2: Correct solution
$(-6)^2 - (-4) = 36 + 4 = 40$

### Answers:

#### 3a: $\boldsymbol{-2 + (5 - (-8)) = 11}$ (or $\boldsymbol{-2 + 5 - (-8) = 11}$)
#### 3b: $\boldsymbol{-4 \div (7 - 9) = 2}$
#### 3c: $\boldsymbol{3 \times (-2) - (-5) = -1}$

#### 5:
- Average change per year: $\boldsymbol{-2}$
- Interpretation: The negative sign means the deer population **decreases** overall.

#### 6a:
- Pattern: Add $3$ each time.
- Next two terms: $\boldsymbol{-2, 1}$

#### 6b:
- Pattern: Subtract $5$ (or add $-5$) each time.
- Next two terms: $\boldsymbol{-11, -16}$

#### 6c:
- Pattern: Multiply by $-2$ each time.
- Next two terms: $\boldsymbol{32, -64}$

#### 7a: $\boldsymbol{-39}$
#### 7b: $\boldsymbol{-44}$
#### 7c: $\boldsymbol{32}$ (assuming $-2(6 + 7b) \times 2$)
#### 7d: $\boldsymbol{192}$

#### 8a:
- Error: Incorrect order of operations (combined $5 - 12$ instead of calculating brackets first).
- Correct solution: $\boldsymbol{5 - 12[4 - (-7)] = -127}$

#### 8b:
- Error: Mistakenly calculated $(-6)^2$ as $-36$ (should be $36$) and misapplied the sign for $-(-4)$.
- Correct solution: $\boldsymbol{(-6)^2 - (-4) = 40}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 3a:

Step1: Analyze operations and brackets

Step2: Apply order of operations (PEMDAS)

Step3: Simplify

Problem 3b:

Step1: Analyze operations

Problem 3c:

Step1: Analyze operations

Problem 5:

Answer:

3a: \(\boldsymbol{-2 + (5 - (-8)) = 11}\) (or \(\boldsymbol{-2 + 5 - (-8) = 11}\))

3b: \(\boldsymbol{-4 \div (7 - 9) = 2}\)

3c: \(\boldsymbol{3 \times (-2) - (-5) = -1}\)

5:

6a:

6b:

6c:

7a: \(\boldsymbol{-39}\)

7b: \(\boldsymbol{-44}\)

7c: \(\boldsymbol{32}\) (assuming \(-2(6 + 7b) \times 2\))

7d: \(\boldsymbol{192}\)

8a:

8b: