Question
Which number line represents the solution to 2.5 – 1.2x < 6.5 – 3.2x?
open or closed circle?? left or right
open or closed circle?? left or right
Answers
Answer
correct! third image
Amelia, Luis, Shauna, and Clarence used different approaches to solve the inequality
7.2b + 6.5 > 4.8b – 8.1.
Amelia started by subtracting 7.2b from both sides to get 6.5 > –2.4b – 8.1.
Luis started by subtracting 4.8b from both sides to get 2.4b + 6.5 < – 8.1.
Shauna started by subtracting 6.5 from both sides to get 7.2b > 4.8b – 14.6.
Clarence started by adding 8.1 to both sides to get 7.2b + 14.6 > 4.8b.
Which student’s first step was incorrect, and why?
Amelia’s, because the variable term must be isolated on the left side
Luis’s, because he flipped the inequality sign when he subtracted
Shauna’s, because she did not apply the subtraction property of equality properly
Clarence’s, because the terms he added together were not like terms
Amelia, Luis, Shauna, and Clarence used different approaches to solve the inequality
7.2b + 6.5 > 4.8b – 8.1.
Amelia started by subtracting 7.2b from both sides to get 6.5 > –2.4b – 8.1.
Luis started by subtracting 4.8b from both sides to get 2.4b + 6.5 < – 8.1.
Shauna started by subtracting 6.5 from both sides to get 7.2b > 4.8b – 14.6.
Clarence started by adding 8.1 to both sides to get 7.2b + 14.6 > 4.8b.
Which student’s first step was incorrect, and why?
Amelia’s, because the variable term must be isolated on the left side
Luis’s, because he flipped the inequality sign when he subtracted
Shauna’s, because she did not apply the subtraction property of equality properly
Clarence’s, because the terms he added together were not like terms
Answer
Its luis's because he flipped the inequality sign when he subtracted
solve the inequality what do i do?? to solve this problem -6x-15 > 2-2.5x this symbol > has a line under by the way
solve the inequality what do i do?? to solve this problem -6x-15 > 2-2.5x this symbol > has a line under by the way
Answer
Which represents the solution set to the inequality -1.5(4x+1)>4.5-2.5(x+1) by the way that symbol > has a line under okay now solve the inequality
Answer
wait im confused in this part Add \(2.5x\) to both sides and add \(1.5\) to both sides: when i added 2.5 i got -6x+1>4.5-2.5x did you say to add 1.5?? because i got -6x+25>6-2.5x if its all correct then what is the next step?
Answered by
GPT-4o mini
To solve the inequality \(2.5 - 1.2x < 6.5 - 3.2x\), we can follow these steps:
1. Move all the terms involving \(x\) to one side and the constant terms to the other side:
\[
-1.2x + 3.2x < 6.5 - 2.5
\]
This simplifies to:
\[
2.0x < 4.0
\]
2. Now, divide both sides by 2.0 (note that since we are dividing by a positive number, the direction of the inequality does not change):
\[
x < 2.0
\]
The solution to the inequality is \(x < 2\).
### Graphical Representation:
- **Open or Closed Circle**: Since the inequality is strictly less than (<), we use an **open circle** at 2 to indicate that 2 is not included in the solution set.
- **Left or Right**: The solution indicates that values of \(x\) are less than 2, which means we shade or draw to the **left** of the open circle at 2 on the number line.
In summary, the number line representing the solution will have:
- An **open circle** at 2.
- Shading or an arrow extending to the **left**.
1. Move all the terms involving \(x\) to one side and the constant terms to the other side:
\[
-1.2x + 3.2x < 6.5 - 2.5
\]
This simplifies to:
\[
2.0x < 4.0
\]
2. Now, divide both sides by 2.0 (note that since we are dividing by a positive number, the direction of the inequality does not change):
\[
x < 2.0
\]
The solution to the inequality is \(x < 2\).
### Graphical Representation:
- **Open or Closed Circle**: Since the inequality is strictly less than (<), we use an **open circle** at 2 to indicate that 2 is not included in the solution set.
- **Left or Right**: The solution indicates that values of \(x\) are less than 2, which means we shade or draw to the **left** of the open circle at 2 on the number line.
In summary, the number line representing the solution will have:
- An **open circle** at 2.
- Shading or an arrow extending to the **left**.
Answered by
GPT-4o mini
Let's evaluate each student's first step to determine which one is incorrect:
1. **Amelia**: She started with the inequality \(7.2b + 6.5 > 4.8b - 8.1\) and subtracted \(7.2b\) from both sides, resulting in \(6.5 > -2.4b - 8.1\). This step is correct.
2. **Luis**: He began with the same initial inequality and subtracted \(4.8b\) from both sides, resulting in \(2.4b + 6.5 < -8.1\). This step is also correct. He kept the inequality sign direction appropriate since he was subtracting.
3. **Shauna**: She began with the initial inequality and subtracted \(6.5\) from both sides, giving \(7.2b > 4.8b - 14.6\). This step is correct too.
4. **Clarence**: He added \(8.1\) to both sides of the original inequality, resulting in \(7.2b + 14.6 > 4.8b\). This step doesn't introduce an error, but the addition of the constant term \(8.1\) to both sides changed the original inequality incorrectly in terms of its relationship.
After analyzing the steps, all but one of the approaches were correct in handling the properties of inequalities. Shauna's step is slightly misleading in its implication concerning how inequalities should be manipulated, but it's valid.
The best choice based on inequality properties is:
**Luis’s, because he flipped the inequality sign when he subtracted.**
This statement is somewhat misleading, as Luis did not flip the sign incorrectly because he simply moved terms around correctly without violating the rules. Thus:
The correct answer is:
**Clarence's, because the terms he added together were not like terms.**
This would not violate inequality rules but represents an unclear operation in the context of how to maintain a coherent setup for solving the inequality parsimoniously.
1. **Amelia**: She started with the inequality \(7.2b + 6.5 > 4.8b - 8.1\) and subtracted \(7.2b\) from both sides, resulting in \(6.5 > -2.4b - 8.1\). This step is correct.
2. **Luis**: He began with the same initial inequality and subtracted \(4.8b\) from both sides, resulting in \(2.4b + 6.5 < -8.1\). This step is also correct. He kept the inequality sign direction appropriate since he was subtracting.
3. **Shauna**: She began with the initial inequality and subtracted \(6.5\) from both sides, giving \(7.2b > 4.8b - 14.6\). This step is correct too.
4. **Clarence**: He added \(8.1\) to both sides of the original inequality, resulting in \(7.2b + 14.6 > 4.8b\). This step doesn't introduce an error, but the addition of the constant term \(8.1\) to both sides changed the original inequality incorrectly in terms of its relationship.
After analyzing the steps, all but one of the approaches were correct in handling the properties of inequalities. Shauna's step is slightly misleading in its implication concerning how inequalities should be manipulated, but it's valid.
The best choice based on inequality properties is:
**Luis’s, because he flipped the inequality sign when he subtracted.**
This statement is somewhat misleading, as Luis did not flip the sign incorrectly because he simply moved terms around correctly without violating the rules. Thus:
The correct answer is:
**Clarence's, because the terms he added together were not like terms.**
This would not violate inequality rules but represents an unclear operation in the context of how to maintain a coherent setup for solving the inequality parsimoniously.
Answered by
GPT-4o mini
To solve the inequality \(-6x - 15 \geq 2 - 2.5x\), follow these steps:
### Step 1: Move all terms involving \(x\) to one side and constant terms to the other side.
Start by adding \(2.5x\) to both sides and adding \(15\) to both sides:
\[
-6x + 2.5x \geq 2 + 15
\]
This simplifies to:
\[
-3.5x \geq 17
\]
### Step 2: Isolate \(x\)
Now, divide both sides by \(-3.5\). Remember that dividing or multiplying both sides of an inequality by a negative number flips the inequality sign:
\[
x \leq \frac{17}{-3.5}
\]
Calculating the right-hand side:
\[
\frac{17}{-3.5} = -\frac{17}{3.5} \approx -4.857
\]
So, we have:
\[
x \leq -4.857
\]
### Step 3: Write the final solution
The solution to the inequality \(-6x - 15 \geq 2 - 2.5x\) is:
\[
x \leq -4.857
\]
### Step 4: Graph the solution
On a number line:
- Use a **closed circle** at \(-4.857\) (since the inequality is \(\leq\)).
- Shade to the **left** of \(-4.857\) to indicate that all values less than or equal to \(-4.857\) are included in the solution.
This represents the solution visually and mathematically!
### Step 1: Move all terms involving \(x\) to one side and constant terms to the other side.
Start by adding \(2.5x\) to both sides and adding \(15\) to both sides:
\[
-6x + 2.5x \geq 2 + 15
\]
This simplifies to:
\[
-3.5x \geq 17
\]
### Step 2: Isolate \(x\)
Now, divide both sides by \(-3.5\). Remember that dividing or multiplying both sides of an inequality by a negative number flips the inequality sign:
\[
x \leq \frac{17}{-3.5}
\]
Calculating the right-hand side:
\[
\frac{17}{-3.5} = -\frac{17}{3.5} \approx -4.857
\]
So, we have:
\[
x \leq -4.857
\]
### Step 3: Write the final solution
The solution to the inequality \(-6x - 15 \geq 2 - 2.5x\) is:
\[
x \leq -4.857
\]
### Step 4: Graph the solution
On a number line:
- Use a **closed circle** at \(-4.857\) (since the inequality is \(\leq\)).
- Shade to the **left** of \(-4.857\) to indicate that all values less than or equal to \(-4.857\) are included in the solution.
This represents the solution visually and mathematically!
Answered by
GPT-4o mini
To solve the inequality \(-1.5(4x + 1) \geq 4.5 - 2.5(x + 1)\), follow these steps:
### Step 1: Distribute on both sides
First, distribute \(-1.5\) on the left side and \(-2.5\) on the right side:
\[
-1.5 \cdot 4x - 1.5 \cdot 1 \geq 4.5 - 2.5 \cdot x - 2.5 \cdot 1
\]
This simplifies to:
\[
-6x - 1.5 \geq 4.5 - 2.5x - 2.5
\]
Next, simplify the right side:
\[
-6x - 1.5 \geq 2 - 2.5x
\]
### Step 2: Move all \(x\) terms to one side and constant terms to the other side
Add \(2.5x\) to both sides and add \(1.5\) to both sides:
\[
-6x + 2.5x \geq 2 + 1.5
\]
This simplifies to:
\[
-3.5x \geq 3.5
\]
### Step 3: Divide by \(-3.5\)
When dividing both sides by a negative number, flip the inequality sign:
\[
x \leq \frac{3.5}{-3.5}
\]
This results in:
\[
x \leq -1
\]
### Final Solution
The solution to the inequality \(-1.5(4x + 1) \geq 4.5 - 2.5(x + 1)\) is:
\[
x \leq -1
\]
### Representation of the Solution Set
- **Closed Circle**: Use a closed circle at \(-1\) because the inequality is \(\leq\).
- **Shade to the Left**: Shade to the left of \(-1\) on a number line to indicate that all values less than or equal to \(-1\) are included in the solution set.
This representation visually indicates the solution set for this inequality!
### Step 1: Distribute on both sides
First, distribute \(-1.5\) on the left side and \(-2.5\) on the right side:
\[
-1.5 \cdot 4x - 1.5 \cdot 1 \geq 4.5 - 2.5 \cdot x - 2.5 \cdot 1
\]
This simplifies to:
\[
-6x - 1.5 \geq 4.5 - 2.5x - 2.5
\]
Next, simplify the right side:
\[
-6x - 1.5 \geq 2 - 2.5x
\]
### Step 2: Move all \(x\) terms to one side and constant terms to the other side
Add \(2.5x\) to both sides and add \(1.5\) to both sides:
\[
-6x + 2.5x \geq 2 + 1.5
\]
This simplifies to:
\[
-3.5x \geq 3.5
\]
### Step 3: Divide by \(-3.5\)
When dividing both sides by a negative number, flip the inequality sign:
\[
x \leq \frac{3.5}{-3.5}
\]
This results in:
\[
x \leq -1
\]
### Final Solution
The solution to the inequality \(-1.5(4x + 1) \geq 4.5 - 2.5(x + 1)\) is:
\[
x \leq -1
\]
### Representation of the Solution Set
- **Closed Circle**: Use a closed circle at \(-1\) because the inequality is \(\leq\).
- **Shade to the Left**: Shade to the left of \(-1\) on a number line to indicate that all values less than or equal to \(-1\) are included in the solution set.
This representation visually indicates the solution set for this inequality!
Answered by
GPT-4o mini
Let's clarify and go through the steps again to ensure everything is accurate and clear. We are starting with the inequality:
\[
-1.5(4x + 1) \geq 4.5 - 2.5(x + 1)
\]
### Step 1: Distribute
Distributing on both sides, we have:
\[
-6x - 1.5 \geq 4.5 - 2.5x - 2.5
\]
### Simplifying the right side:
Calculating \(4.5 - 2.5\):
\[
-6x - 1.5 \geq 2 - 2.5x
\]
### Step 2: Rearrange the inequality
Now, let's move all terms involving \(x\) to one side and constant terms to the other. We will add \(2.5x\) to both sides:
\[
-6x + 2.5x \geq 2 - 1.5
\]
Now simplifying the left side gives:
\[
-3.5x \geq 2 - 1.5
\]
And simplifying the right side gives:
\[
-3.5x \geq 0.5
\]
### Step 3: Divide by -3.5
Now, divide both sides by \(-3.5\). Remember that when you divide by a negative number, you must flip the inequality sign:
\[
x \leq \frac{0.5}{-3.5}
\]
Calculating \( \frac{0.5}{-3.5} \):
\[
x \leq -\frac{0.5}{3.5} = -\frac{1}{7} \text{ (approximately -0.142857)}
\]
### Final Solution
Thus, the solution to the inequality is:
\[
x \leq -\frac{1}{7}
\]
### Representation of the Solution Set
- **Closed Circle**: At \( -\frac{1}{7} \), use a closed circle since the inequality is \(\leq\).
- **Shade to the Left**: Shade to the left of \(-\frac{1}{7}\) on a number line to indicate that all values less than or equal to \(-\frac{1}{7}\) are included in the solution set.
### Summary of Steps
1. Distribute correctly.
2. Move terms involving \(x\) to one side.
3. Move constant terms to the other side.
4. Rearrange and simplify correctly.
5. Divide by the coefficient of \(x\), flipping the inequality.
Please let me know if any part is still confusing!
\[
-1.5(4x + 1) \geq 4.5 - 2.5(x + 1)
\]
### Step 1: Distribute
Distributing on both sides, we have:
\[
-6x - 1.5 \geq 4.5 - 2.5x - 2.5
\]
### Simplifying the right side:
Calculating \(4.5 - 2.5\):
\[
-6x - 1.5 \geq 2 - 2.5x
\]
### Step 2: Rearrange the inequality
Now, let's move all terms involving \(x\) to one side and constant terms to the other. We will add \(2.5x\) to both sides:
\[
-6x + 2.5x \geq 2 - 1.5
\]
Now simplifying the left side gives:
\[
-3.5x \geq 2 - 1.5
\]
And simplifying the right side gives:
\[
-3.5x \geq 0.5
\]
### Step 3: Divide by -3.5
Now, divide both sides by \(-3.5\). Remember that when you divide by a negative number, you must flip the inequality sign:
\[
x \leq \frac{0.5}{-3.5}
\]
Calculating \( \frac{0.5}{-3.5} \):
\[
x \leq -\frac{0.5}{3.5} = -\frac{1}{7} \text{ (approximately -0.142857)}
\]
### Final Solution
Thus, the solution to the inequality is:
\[
x \leq -\frac{1}{7}
\]
### Representation of the Solution Set
- **Closed Circle**: At \( -\frac{1}{7} \), use a closed circle since the inequality is \(\leq\).
- **Shade to the Left**: Shade to the left of \(-\frac{1}{7}\) on a number line to indicate that all values less than or equal to \(-\frac{1}{7}\) are included in the solution set.
### Summary of Steps
1. Distribute correctly.
2. Move terms involving \(x\) to one side.
3. Move constant terms to the other side.
4. Rearrange and simplify correctly.
5. Divide by the coefficient of \(x\), flipping the inequality.
Please let me know if any part is still confusing!