Question
Image to Text
Graph the solution of this inequality:
[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Drag a point to the number line.
Summary
The image presents an inequality problem: (\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3). Below the inequality, there is a number line ranging from -7 to 7. There are also draggable points with arrows indicating possible directions, suggesting that the user can place these points on the number line to graph the solution of the inequality.
Graph the solution of this inequality:
[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Drag a point to the number line.
Summary
The image presents an inequality problem: (\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3). Below the inequality, there is a number line ranging from -7 to 7. There are also draggable points with arrows indicating possible directions, suggesting that the user can place these points on the number line to graph the solution of the inequality.
Answers
Answer
CLOSED POINT GOING THIS WAY <---- ON 2 NOT -2 THAnK YOU AI now What is the solution set for the inequality?
(–∞, 2]
(–∞, ]
[2, ∞)
[, ∞)
(–∞, 2]
(–∞, ]
[2, ∞)
[, ∞)
Answer
HOORAY!!
Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?
x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)
Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?
x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)
Answered by
GPT-4o mini
To solve the inequality
\[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3,
\]
we first simplify the left side:
1. Distribute \(\frac{3}{7}\):
\[
\frac{3}{7} \times 35x - \frac{3}{7} \times 14 = 15x - 6.
\]
Thus, the left side simplifies to \(15x - 6\).
2. Rewrite the inequality:
\[
15x - 6 \leq \frac{21x}{2} + 3.
\]
3. Next, we need to eliminate the fraction on the right side. We can multiply the entire inequality by 2 (which does not change the inequality sign as we are multiplying by a positive number):
\[
2(15x - 6) \leq 21x + 6.
\]
This simplifies to:
\[
30x - 12 \leq 21x + 6.
\]
4. Now, we isolate \(x\). Subtract \(21x\) from both sides:
\[
30x - 21x - 12 \leq 6,
\]
which simplifies to:
\[
9x - 12 \leq 6.
\]
5. Then, add 12 to both sides:
\[
9x \leq 18.
\]
6. Finally, divide both sides by 9:
\[
x \leq 2.
\]
So, the solution to the inequality is \(x \leq 2\).
### Graphing the solution
To graph the solution on the number line from -7 to 7:
- Draw a closed circle at \(2\) (indicating that \(2\) is included in the solution).
- Shade the region to the left of \(2\) to indicate all values less than or equal to \(2\).
### Final Step
You can drag a point to the number line at \(2\) and shade to the left to graph the solution \(x \leq 2\).
\[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3,
\]
we first simplify the left side:
1. Distribute \(\frac{3}{7}\):
\[
\frac{3}{7} \times 35x - \frac{3}{7} \times 14 = 15x - 6.
\]
Thus, the left side simplifies to \(15x - 6\).
2. Rewrite the inequality:
\[
15x - 6 \leq \frac{21x}{2} + 3.
\]
3. Next, we need to eliminate the fraction on the right side. We can multiply the entire inequality by 2 (which does not change the inequality sign as we are multiplying by a positive number):
\[
2(15x - 6) \leq 21x + 6.
\]
This simplifies to:
\[
30x - 12 \leq 21x + 6.
\]
4. Now, we isolate \(x\). Subtract \(21x\) from both sides:
\[
30x - 21x - 12 \leq 6,
\]
which simplifies to:
\[
9x - 12 \leq 6.
\]
5. Then, add 12 to both sides:
\[
9x \leq 18.
\]
6. Finally, divide both sides by 9:
\[
x \leq 2.
\]
So, the solution to the inequality is \(x \leq 2\).
### Graphing the solution
To graph the solution on the number line from -7 to 7:
- Draw a closed circle at \(2\) (indicating that \(2\) is included in the solution).
- Shade the region to the left of \(2\) to indicate all values less than or equal to \(2\).
### Final Step
You can drag a point to the number line at \(2\) and shade to the left to graph the solution \(x \leq 2\).
Answered by
GPT-4o mini
The solution set for the inequality \(x \leq 2\) is expressed in interval notation as:
\[
(-\infty, 2]
\]
This means that \(x\) can take any value from negative infinity up to and including \(2\).
So the correct choice from your options is:
**(–∞, 2]**
\[
(-\infty, 2]
\]
This means that \(x\) can take any value from negative infinity up to and including \(2\).
So the correct choice from your options is:
**(–∞, 2]**
Answered by
GPT-4o mini
To solve the inequality
\[
5.1(3 + 2.2x) > -14.25 - 6(1.7x + 4),
\]
let's break it down step by step.
1. **Distribute on both sides:**
**Left side:**
\[
5.1 \times 3 + 5.1 \times 2.2x = 15.3 + 11.22x.
\]
**Right side:**
\[
-14.25 - 6(1.7x + 4) = -14.25 - 10.2x - 24 = -14.25 - 10.2x - 24 = -38.25 - 10.2x.
\]
So, rewriting the inequality gives us:
\[
15.3 + 11.22x > -38.25 - 10.2x.
\]
2. **Combine like terms:**
First, add \(10.2x\) to both sides:
\[
15.3 + 11.22x + 10.2x > -38.25.
\]
This simplifies to:
\[
15.3 + 21.42x > -38.25.
\]
3. **Now, isolate the term with \(x\):**
Subtract \(15.3\) from both sides:
\[
21.42x > -38.25 - 15.3.
\]
This evaluates to:
\[
21.42x > -53.55.
\]
4. **Divide both sides by \(21.42\):**
\[
x > \frac{-53.55}{21.42}.
\]
Calculating the right-hand side gives:
\[
x > -2.5.
\]
### Conclusion:
The solution of the inequality is:
\[
x > -2.5.
\]
In interval notation, the solution set is:
\[
(-2.5, \infty).
\]
So, the correct answer choice from your options is:
**(–2.5, ∞)**.
\[
5.1(3 + 2.2x) > -14.25 - 6(1.7x + 4),
\]
let's break it down step by step.
1. **Distribute on both sides:**
**Left side:**
\[
5.1 \times 3 + 5.1 \times 2.2x = 15.3 + 11.22x.
\]
**Right side:**
\[
-14.25 - 6(1.7x + 4) = -14.25 - 10.2x - 24 = -14.25 - 10.2x - 24 = -38.25 - 10.2x.
\]
So, rewriting the inequality gives us:
\[
15.3 + 11.22x > -38.25 - 10.2x.
\]
2. **Combine like terms:**
First, add \(10.2x\) to both sides:
\[
15.3 + 11.22x + 10.2x > -38.25.
\]
This simplifies to:
\[
15.3 + 21.42x > -38.25.
\]
3. **Now, isolate the term with \(x\):**
Subtract \(15.3\) from both sides:
\[
21.42x > -38.25 - 15.3.
\]
This evaluates to:
\[
21.42x > -53.55.
\]
4. **Divide both sides by \(21.42\):**
\[
x > \frac{-53.55}{21.42}.
\]
Calculating the right-hand side gives:
\[
x > -2.5.
\]
### Conclusion:
The solution of the inequality is:
\[
x > -2.5.
\]
In interval notation, the solution set is:
\[
(-2.5, \infty).
\]
So, the correct answer choice from your options is:
**(–2.5, ∞)**.
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