Question
Which of the x values are solutions to the inequality 4(2 – x) > –2x – 3(4x + 1)? Check all that apply.
x = –1.1
x = –2.2
x = 0
x = –10
x = 10
x = –1.1
x = –2.2
x = 0
x = –10
x = 10
Answers
Answer
The inequality given is ( \frac{3}{4}(x + 8) > \frac{1}{2}(x + 10) ). You need to solve this inequality and then graph the solution on the number line provided, which ranges from -5 to 5. open or close circle? left or right
Answer
incorrect
Answer
its genuinely uncorrect i put open circle arrow going right on -4 and its wrong?
Answer
You need to solve this inequality and then graph the solution on the number line provided, which ranges from -5 to 5. its 3/4(x+8)>1/2(2x+10) open or closed circle? left or right
Answer
no no my mistake im sorry thank you tho
What is the solution to 3/4(x+8)>1/2(2x+10)?
(–∞, –4)
(–4, ∞)
(–∞, 4)
(4, ∞)
What is the solution to 3/4(x+8)>1/2(2x+10)?
(–∞, –4)
(–4, ∞)
(–∞, 4)
(4, ∞)
Answered by
GPT-4o mini
To solve the inequality \( 4(2 - x) > -2x - 3(4x + 1) \), let's start by simplifying both sides of the inequality.
1. Distribute the \( 4 \) on the left side:
\[
4(2 - x) = 8 - 4x
\]
2. Distribute the \( -3 \) on the right side:
\[
-3(4x + 1) = -12x - 3
\]
So, the right side becomes:
\[
-2x - 12x - 3 = -14x - 3
\]
Thus, the inequality simplifies to:
\[
8 - 4x > -14x - 3
\]
3. Next, let's add \( 14x \) to both sides:
\[
8 - 4x + 14x > -3
\]
Simplifying gives:
\[
8 + 10x > -3
\]
4. Now, subtract \( 8 \) from both sides:
\[
10x > -3 - 8
\]
So, we have:
\[
10x > -11
\]
5. Finally, divide by \( 10 \):
\[
x > -\frac{11}{10}
\]
or
\[
x > -1.1
\]
Now that we have the solution \( x > -1.1 \), we can check the given x values:
- \( x = -1.1 \): **Not a solution** (since it is not greater than -1.1)
- \( x = -2.2 \): **Not a solution** (since -2.2 is less than -1.1)
- \( x = 0 \): **Solution** (since 0 is greater than -1.1)
- \( x = -10 \): **Not a solution** (since -10 is less than -1.1)
- \( x = 10 \): **Solution** (since 10 is greater than -1.1)
Thus, the values that are solutions to the inequality are:
- **x = 0**
- **x = 10**
1. Distribute the \( 4 \) on the left side:
\[
4(2 - x) = 8 - 4x
\]
2. Distribute the \( -3 \) on the right side:
\[
-3(4x + 1) = -12x - 3
\]
So, the right side becomes:
\[
-2x - 12x - 3 = -14x - 3
\]
Thus, the inequality simplifies to:
\[
8 - 4x > -14x - 3
\]
3. Next, let's add \( 14x \) to both sides:
\[
8 - 4x + 14x > -3
\]
Simplifying gives:
\[
8 + 10x > -3
\]
4. Now, subtract \( 8 \) from both sides:
\[
10x > -3 - 8
\]
So, we have:
\[
10x > -11
\]
5. Finally, divide by \( 10 \):
\[
x > -\frac{11}{10}
\]
or
\[
x > -1.1
\]
Now that we have the solution \( x > -1.1 \), we can check the given x values:
- \( x = -1.1 \): **Not a solution** (since it is not greater than -1.1)
- \( x = -2.2 \): **Not a solution** (since -2.2 is less than -1.1)
- \( x = 0 \): **Solution** (since 0 is greater than -1.1)
- \( x = -10 \): **Not a solution** (since -10 is less than -1.1)
- \( x = 10 \): **Solution** (since 10 is greater than -1.1)
Thus, the values that are solutions to the inequality are:
- **x = 0**
- **x = 10**
Answered by
GPT-4o mini
To solve the inequality
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10),
\]
let's start by clearing the fractions.
1. Multiply both sides by 4 to eliminate the denominators:
\[
4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
2. Distribute on both sides:
\[
3x + 24 > 2x + 20.
\]
3. Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20,
\]
simplifying to:
\[
x + 24 > 20.
\]
4. Now, subtract \(24\) from both sides:
\[
x > 20 - 24,
\]
which simplifies to:
\[
x > -4.
\]
### Graphing the Solution
The solution \(x > -4\) means that \(x\) can take any value greater than \(-4\), but not equal to \(-4\).
1. **Open Circle**: Since \(-4\) is not included in the solution (the inequality is strict), you would use an **open circle** at \(-4\).
2. **Arrow to the Right**: This indicates that the solution continues to the right (greater than \(-4\)).
### Number Line
On a number line ranging from \(-5\) to \(5\):
- Place an open circle at \(-4\).
- Shade or draw an arrow to the right from \(-4\) to indicate all values greater than \(-4\).
Here's a representation in text form:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
(====================>
```
In conclusion, the solution is \(x > -4\) with an **open circle** at \(-4\) and shading to the **right**.
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10),
\]
let's start by clearing the fractions.
1. Multiply both sides by 4 to eliminate the denominators:
\[
4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
2. Distribute on both sides:
\[
3x + 24 > 2x + 20.
\]
3. Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20,
\]
simplifying to:
\[
x + 24 > 20.
\]
4. Now, subtract \(24\) from both sides:
\[
x > 20 - 24,
\]
which simplifies to:
\[
x > -4.
\]
### Graphing the Solution
The solution \(x > -4\) means that \(x\) can take any value greater than \(-4\), but not equal to \(-4\).
1. **Open Circle**: Since \(-4\) is not included in the solution (the inequality is strict), you would use an **open circle** at \(-4\).
2. **Arrow to the Right**: This indicates that the solution continues to the right (greater than \(-4\)).
### Number Line
On a number line ranging from \(-5\) to \(5\):
- Place an open circle at \(-4\).
- Shade or draw an arrow to the right from \(-4\) to indicate all values greater than \(-4\).
Here's a representation in text form:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
(====================>
```
In conclusion, the solution is \(x > -4\) with an **open circle** at \(-4\) and shading to the **right**.
Answered by
GPT-4o mini
Let's go through the inequality step by step again to ensure we get it correct:
The original inequality is:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 (the least common multiple of the denominators) to eliminate the fractions:
\[
4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
### Step 2: Distribute
Distributing on both sides gives us:
\[
3x + 24 > 2x + 20.
\]
### Step 3: Move all terms involving \(x\) to one side
Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20.
\]
This simplifies to:
\[
x + 24 > 20.
\]
### Step 4: Isolate \(x\)
Subtract 24 from both sides:
\[
x > 20 - 24,
\]
which simplifies to:
\[
x > -4.
\]
### Conclusion of the solution
The solution to the inequality is \(x > -4\).
### Graph on the number line
- **Open Circle** at \(-4\): indicates that \(-4\) is not included in the solution.
- **Arrow to the Right**: indicates that the solution includes all numbers greater than \(-4\).
In a basic number line representation between \(-5\) and \(5\):
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
(====================>
```
### Summary
- **Open circle** at \(-4\) (because \(-4\) is not part of the solution)
- Shade or draw an **arrow to the right** indicating all values greater than \(-4\).
If you still consider this incorrect, please clarify which particular part seems off, and I’ll do my best to address it!
The original inequality is:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 (the least common multiple of the denominators) to eliminate the fractions:
\[
4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
### Step 2: Distribute
Distributing on both sides gives us:
\[
3x + 24 > 2x + 20.
\]
### Step 3: Move all terms involving \(x\) to one side
Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20.
\]
This simplifies to:
\[
x + 24 > 20.
\]
### Step 4: Isolate \(x\)
Subtract 24 from both sides:
\[
x > 20 - 24,
\]
which simplifies to:
\[
x > -4.
\]
### Conclusion of the solution
The solution to the inequality is \(x > -4\).
### Graph on the number line
- **Open Circle** at \(-4\): indicates that \(-4\) is not included in the solution.
- **Arrow to the Right**: indicates that the solution includes all numbers greater than \(-4\).
In a basic number line representation between \(-5\) and \(5\):
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
(====================>
```
### Summary
- **Open circle** at \(-4\) (because \(-4\) is not part of the solution)
- Shade or draw an **arrow to the right** indicating all values greater than \(-4\).
If you still consider this incorrect, please clarify which particular part seems off, and I’ll do my best to address it!
Answered by
GPT-4o mini
I apologize for the confusion earlier. Let's re-evaluate the inequality clearly step-by-step and ensure that we understand the graphing part correctly.
The inequality we are solving is:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10).
\]
### Step 1: Clear the fractions
To eliminate the fractions, we can multiply both sides by 4:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
### Step 2: Distribute
Now distribute on both sides:
\[
3x + 24 > 2x + 20.
\]
### Step 3: Isolate \(x\)
Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20 \implies x + 24 > 20.
\]
Now subtract 24 from both sides:
\[
x > 20 - 24 \implies x > -4.
\]
### Interpretation
The solution is \(x > -4\).
### Graphing the Solution
1. **Open Circle at -4**: Since the inequality is strict (greater than, not greater than or equal to), we use an **open circle** at \(-4\).
2. **Arrow to the Right**: This indicates that values greater than \(-4\) are included in the solution.
### Number Line Representation
On a number line from -5 to 5:
- Draw an **open circle** at \(-4\).
- Shade or draw an **arrow pointing right** from \(-4\) to indicate all values greater than \(-4\).
Here’s how it looks:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
O====================>
```
The **O** indicates the open circle at \(-4\).
### Conclusion
- The solution \(x > -4\) is correct.
- You should indeed use an **open circle** at \(-4\) and shade to the **right**.
If there is still a misunderstanding or a specific aspect of the graph that seems incorrect, please let me know! Thank you for your patience.
The inequality we are solving is:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(x + 10).
\]
### Step 1: Clear the fractions
To eliminate the fractions, we can multiply both sides by 4:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(x + 10).
\]
### Step 2: Distribute
Now distribute on both sides:
\[
3x + 24 > 2x + 20.
\]
### Step 3: Isolate \(x\)
Subtract \(2x\) from both sides:
\[
3x - 2x + 24 > 20 \implies x + 24 > 20.
\]
Now subtract 24 from both sides:
\[
x > 20 - 24 \implies x > -4.
\]
### Interpretation
The solution is \(x > -4\).
### Graphing the Solution
1. **Open Circle at -4**: Since the inequality is strict (greater than, not greater than or equal to), we use an **open circle** at \(-4\).
2. **Arrow to the Right**: This indicates that values greater than \(-4\) are included in the solution.
### Number Line Representation
On a number line from -5 to 5:
- Draw an **open circle** at \(-4\).
- Shade or draw an **arrow pointing right** from \(-4\) to indicate all values greater than \(-4\).
Here’s how it looks:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1
O====================>
```
The **O** indicates the open circle at \(-4\).
### Conclusion
- The solution \(x > -4\) is correct.
- You should indeed use an **open circle** at \(-4\) and shade to the **right**.
If there is still a misunderstanding or a specific aspect of the graph that seems incorrect, please let me know! Thank you for your patience.
Answered by
GPT-4o mini
Thank you for your patience, and I appreciate the opportunity to clarify everything. Let’s solve the correct inequality you just provided:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 to eliminate the denominators:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(2x + 10).
\]
### Step 2: Distribute both sides
Distributing on both sides gives:
\[
3x + 24 > 4x + 20.
\]
### Step 3: Isolate \(x\)
Now, we'll isolate \(x\) by moving all terms involving \(x\) to one side and constant terms to the other side:
1. Subtract \(3x\) from both sides:
\[
24 > x + 20.
\]
2. Next, subtract \(20\) from both sides:
\[
24 - 20 > x \implies 4 > x.
\]
This can also be written as:
\[
x < 4.
\]
### Summary of the Solution
The solution to the inequality is:
\[
x < 4.
\]
### Step 4: Graphing on the number line
- **Open Circle** at \(4\): Since the inequality is strict (<), you would place an **open circle** at \(4\).
- **Arrow to the Left**: This indicates that \(x\) can take any value less than \(4\).
### Number Line Representation
On a number line that ranges from -5 to 5, it would look like this:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1 2 3 4 5
O<===================
```
Where **O** represents the open circle at \(4\) and the shading to the left represents all values less than \(4\).
### Conclusion
- Use an **open circle** at \(4\).
- Shade or draw an **arrow to the left** indicating all values less than \(4\).
Thank you for bringing this to my attention. If you have any further questions, feel free to ask!
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 to eliminate the denominators:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(2x + 10).
\]
### Step 2: Distribute both sides
Distributing on both sides gives:
\[
3x + 24 > 4x + 20.
\]
### Step 3: Isolate \(x\)
Now, we'll isolate \(x\) by moving all terms involving \(x\) to one side and constant terms to the other side:
1. Subtract \(3x\) from both sides:
\[
24 > x + 20.
\]
2. Next, subtract \(20\) from both sides:
\[
24 - 20 > x \implies 4 > x.
\]
This can also be written as:
\[
x < 4.
\]
### Summary of the Solution
The solution to the inequality is:
\[
x < 4.
\]
### Step 4: Graphing on the number line
- **Open Circle** at \(4\): Since the inequality is strict (<), you would place an **open circle** at \(4\).
- **Arrow to the Left**: This indicates that \(x\) can take any value less than \(4\).
### Number Line Representation
On a number line that ranges from -5 to 5, it would look like this:
```
---|----|----|----|----|----|----|----
-5 -4 -3 -2 -1 0 1 2 3 4 5
O<===================
```
Where **O** represents the open circle at \(4\) and the shading to the left represents all values less than \(4\).
### Conclusion
- Use an **open circle** at \(4\).
- Shade or draw an **arrow to the left** indicating all values less than \(4\).
Thank you for bringing this to my attention. If you have any further questions, feel free to ask!
Answered by
GPT-4o mini
No problem at all! Let's solve the inequality step by step to determine its solution.
Given the inequality:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 to eliminate the fractions:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(2x + 10).
\]
### Step 2: Distribute both sides
Now distribute on both sides:
\[
3x + 24 > 4x + 20.
\]
### Step 3: Isolate \(x\)
Now we'll move terms involving \(x\) to one side and constant terms to the other side:
1. Subtract \(3x\) from both sides:
\[
24 > x + 20.
\]
2. Subtract \(20\) from both sides:
\[
24 - 20 > x \implies 4 > x.
\]
This can also be expressed as:
\[
x < 4.
\]
### Final Solution
The solution is \(x < 4\), which operates over the interval:
\[
(-\infty, 4).
\]
### Answer
Thus, among the options provided, the correct answer is:
\[
\text{(–∞, 4)}.
\]
If you have any further questions or need additional clarification, feel free to ask!
Given the inequality:
\[
\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10).
\]
### Step 1: Clear the fractions
Multiply both sides by 4 to eliminate the fractions:
\[
4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10).
\]
This simplifies to:
\[
3(x + 8) > 2(2x + 10).
\]
### Step 2: Distribute both sides
Now distribute on both sides:
\[
3x + 24 > 4x + 20.
\]
### Step 3: Isolate \(x\)
Now we'll move terms involving \(x\) to one side and constant terms to the other side:
1. Subtract \(3x\) from both sides:
\[
24 > x + 20.
\]
2. Subtract \(20\) from both sides:
\[
24 - 20 > x \implies 4 > x.
\]
This can also be expressed as:
\[
x < 4.
\]
### Final Solution
The solution is \(x < 4\), which operates over the interval:
\[
(-\infty, 4).
\]
### Answer
Thus, among the options provided, the correct answer is:
\[
\text{(–∞, 4)}.
\]
If you have any further questions or need additional clarification, feel free to ask!
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