Answer :
Sure, I can provide detailed answers for each part of the question.
### 1. Which inequality represents the sentence?
The product of a number [tex]\(a\)[/tex] and 6 is no less than 14.
This means that 6 times [tex]\(a\)[/tex] is at least 14.
This translates to the inequality:
[tex]\[ 6a \geq 14 \][/tex]
### 2. Which inequality represents the graph?
There are multiple inequalities given. We need to identify which one is correct based on the options:
#### Inequality Options:
- [tex]\(6a \leq 14\)[/tex]
- [tex]\(-2 \leq n < 1\)[/tex]
- [tex]\(6a > 14\)[/tex]
- [tex]\( -2 < n < 1 \)[/tex]
- [tex]\( 6a < 14 \)[/tex]
- [tex]\( n \leq -2 \)[/tex] or [tex]\( n \geq 1 \)[/tex]
- [tex]\( n \leq -2 \)[/tex] or [tex]\( n > 1 \)[/tex]
Since there is no specific graph provided here, we cannot definitively choose based on visual representation. However, if we consider the logical pair to the first inequality (provided in question 1), the complementary inequality should be:
[tex]\[ 6a \leq 14 \][/tex]
### 3. Does the graph represent a function?
Given the domain and range constraints:
Domain:
[tex]\[ -1 \leq x \leq 2 \][/tex]
Range:
[tex]\[ -1 \leq y \leq 2 \][/tex]
To check if the graph represents a function:
- A graph represents a function if any vertical line drawn through the domain intersects the graph at most once.
Based on the given domain and range constraints, the graph does represent a function within these confined limits:
(a) does represent a function.
### 4. What is the solution of the equation?
Solve the equation:
[tex]\[ \frac{1}{2}(8x + 4) = 6(x + 2) \][/tex]
Let's solve it step-by-step:
1. Expand both sides of the equation:
[tex]\[ \frac{1}{2}(8x + 4) = 4x + 2 \][/tex]
[tex]\[ 6(x + 2) = 6x + 12 \][/tex]
2. Substitute the expanded forms back into the equation:
[tex]\[ 4x + 2 = 6x + 12 \][/tex]
3. Move all the terms involving [tex]\(x\)[/tex] to one side and the constants to the other:
[tex]\[ 4x + 2 - 6x = 12 \][/tex]
[tex]\[ -2x + 2 = 12 \][/tex]
4. Isolate [tex]\(x\)[/tex]:
[tex]\[ -2x = 10 \][/tex]
[tex]\[ x = -5 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = -5 \][/tex]
### 1. Which inequality represents the sentence?
The product of a number [tex]\(a\)[/tex] and 6 is no less than 14.
This means that 6 times [tex]\(a\)[/tex] is at least 14.
This translates to the inequality:
[tex]\[ 6a \geq 14 \][/tex]
### 2. Which inequality represents the graph?
There are multiple inequalities given. We need to identify which one is correct based on the options:
#### Inequality Options:
- [tex]\(6a \leq 14\)[/tex]
- [tex]\(-2 \leq n < 1\)[/tex]
- [tex]\(6a > 14\)[/tex]
- [tex]\( -2 < n < 1 \)[/tex]
- [tex]\( 6a < 14 \)[/tex]
- [tex]\( n \leq -2 \)[/tex] or [tex]\( n \geq 1 \)[/tex]
- [tex]\( n \leq -2 \)[/tex] or [tex]\( n > 1 \)[/tex]
Since there is no specific graph provided here, we cannot definitively choose based on visual representation. However, if we consider the logical pair to the first inequality (provided in question 1), the complementary inequality should be:
[tex]\[ 6a \leq 14 \][/tex]
### 3. Does the graph represent a function?
Given the domain and range constraints:
Domain:
[tex]\[ -1 \leq x \leq 2 \][/tex]
Range:
[tex]\[ -1 \leq y \leq 2 \][/tex]
To check if the graph represents a function:
- A graph represents a function if any vertical line drawn through the domain intersects the graph at most once.
Based on the given domain and range constraints, the graph does represent a function within these confined limits:
(a) does represent a function.
### 4. What is the solution of the equation?
Solve the equation:
[tex]\[ \frac{1}{2}(8x + 4) = 6(x + 2) \][/tex]
Let's solve it step-by-step:
1. Expand both sides of the equation:
[tex]\[ \frac{1}{2}(8x + 4) = 4x + 2 \][/tex]
[tex]\[ 6(x + 2) = 6x + 12 \][/tex]
2. Substitute the expanded forms back into the equation:
[tex]\[ 4x + 2 = 6x + 12 \][/tex]
3. Move all the terms involving [tex]\(x\)[/tex] to one side and the constants to the other:
[tex]\[ 4x + 2 - 6x = 12 \][/tex]
[tex]\[ -2x + 2 = 12 \][/tex]
4. Isolate [tex]\(x\)[/tex]:
[tex]\[ -2x = 10 \][/tex]
[tex]\[ x = -5 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = -5 \][/tex]
