Question
Which graph represents the solution set for the inequality StartFraction one-half EndFraction x is less than or equal to 18.x ≤ 18?
0 1 2 3 4 5 6 7 8 9 10
0 6 12 18 24 30 36 42 48 54 60
0 1 2 3 4 5 6 7 8 9 10
0 6 12 18 24 30 36 42 48 54 60
Summary
The image shows four number lines with arrows and orange dots. The first and third number lines are labeled from 0 to 10, with an orange dot at 9 on the first and 7 on the third. The second and fourth number lines are labeled from 0 to 60 in increments of 6, with an orange dot at 36 on both. Each number line has a circular checkbox to the left, presumably for selection purposes.
0 1 2 3 4 5 6 7 8 9 10
0 6 12 18 24 30 36 42 48 54 60
0 1 2 3 4 5 6 7 8 9 10
0 6 12 18 24 30 36 42 48 54 60
Summary
The image shows four number lines with arrows and orange dots. The first and third number lines are labeled from 0 to 10, with an orange dot at 9 on the first and 7 on the third. The second and fourth number lines are labeled from 0 to 60 in increments of 6, with an orange dot at 36 on both. Each number line has a circular checkbox to the left, presumably for selection purposes.
Answers
Answer
its 0 to 36 aka 4th number line
Match the solution set given in inequality notation with the solution set given in interval notation.
x ≤ 7.8 --------> (−∞, 7.8]
x < 7.8 --------> (−∞, 7.8)
x > 7.8 --------> (7.8, ∞)
x ≥ 7.8 --------> [7.8, ∞)
Summary
The image shows a matching task where solution sets given in inequality notation need to be matched with their corresponding interval notation. The inequalities and their corresponding interval notations are as follows:
( x \leq 7.8 ) matches ( (−∞, 7.8] )
( x < 7.8 ) matches ( (−∞, 7.8) )
( x > 7.8 ) matches ( (7.8, ∞) )
( x \geq 7.8 ) matches ( [7.8, ∞) )
Match the solution set given in inequality notation with the solution set given in interval notation.
x ≤ 7.8 --------> (−∞, 7.8]
x < 7.8 --------> (−∞, 7.8)
x > 7.8 --------> (7.8, ∞)
x ≥ 7.8 --------> [7.8, ∞)
Summary
The image shows a matching task where solution sets given in inequality notation need to be matched with their corresponding interval notation. The inequalities and their corresponding interval notations are as follows:
( x \leq 7.8 ) matches ( (−∞, 7.8] )
( x < 7.8 ) matches ( (−∞, 7.8) )
( x > 7.8 ) matches ( (7.8, ∞) )
( x \geq 7.8 ) matches ( [7.8, ∞) )
Answer
CORRECT guys its legit just across from each other
Image to Text
Graph the solution of this inequality:
[
\frac{4}{9}x - 10 > \frac{x}{3} - 12
]
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0
Drag a point to the number line.
Summary
The image contains a mathematical inequality (\frac{4}{9}x - 10 > \frac{x}{3} - 12) that needs to be solved and graphed on a number line. The number line ranges from -20 to 0. There are orange points below the number line that can be dragged to represent the solution. The task is to drag these points to the correct positions on the number line to graph the solution of the inequality.
Image to Text
Graph the solution of this inequality:
[
\frac{4}{9}x - 10 > \frac{x}{3} - 12
]
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0
Drag a point to the number line.
Summary
The image contains a mathematical inequality (\frac{4}{9}x - 10 > \frac{x}{3} - 12) that needs to be solved and graphed on a number line. The number line ranges from -20 to 0. There are orange points below the number line that can be dragged to represent the solution. The task is to drag these points to the correct positions on the number line to graph the solution of the inequality.
Answer
hey guys so i clicked on example it did something by itself but YES ai is correct you do open circle arrow is moving to the right -->> and place it on -18
but now What is the solution to x – 10 > – 12?
(–∞, –18)
(–18, ∞)
(–∞, 18)
(18, ∞)
but now What is the solution to x – 10 > – 12?
(–∞, –18)
(–18, ∞)
(–∞, 18)
(18, ∞)
Answer
What is the solution to 4/9 x – 10 > x/3 – 12?
(–∞, –18)
(–18, ∞)
(–∞, 18)
(18, ∞)
(–∞, –18)
(–18, ∞)
(–∞, 18)
(18, ∞)
Answer
To make a profit, a company’s revenue must be greater than its operating costs. The company’s revenue is modeled by the expression 7.5x – 100, where x represents the number of items sold. The company’s operation costs are modeled by the expression 79.86 + 5.8x. How many items does the company need to sell to make a profit?
The inequality that will determine the number of items that need to be sold to make a profit is
.
The solution to the inequality is
.
The company must sell at least
items to make a profit.
The inequality that will determine the number of items that need to be sold to make a profit is
.
The solution to the inequality is
.
The company must sell at least
items to make a profit.
Answered by
GPT-4o mini
To solve the inequality \( \frac{1}{2} x \leq 18 \), we first eliminate the fraction by multiplying both sides of the inequality by 2:
\[
x \leq 36
\]
This means that the solution set consists of all values of \( x \) that are less than or equal to 36.
Now, let's analyze the number lines you described:
1. The first number line is from 0 to 10, with a dot at 9.
2. The second number line is from 0 to 60, with a dot at 36.
3. The third number line is from 0 to 10, with a dot at 7.
4. The fourth number line is also from 0 to 60, with a dot at 36.
Since the solution set of the inequality \( x \leq 36 \) includes all values up to and including 36, the number lines that include a dot at 36 correctly represent the solution.
Both the second and fourth number lines are labeled from 0 to 60 with a dot at 36, which makes them the correct representations of the solution set for the inequality.
\[
x \leq 36
\]
This means that the solution set consists of all values of \( x \) that are less than or equal to 36.
Now, let's analyze the number lines you described:
1. The first number line is from 0 to 10, with a dot at 9.
2. The second number line is from 0 to 60, with a dot at 36.
3. The third number line is from 0 to 10, with a dot at 7.
4. The fourth number line is also from 0 to 60, with a dot at 36.
Since the solution set of the inequality \( x \leq 36 \) includes all values up to and including 36, the number lines that include a dot at 36 correctly represent the solution.
Both the second and fourth number lines are labeled from 0 to 60 with a dot at 36, which makes them the correct representations of the solution set for the inequality.
Answered by
GPT-4o mini
Let's match the solution sets given in inequality notation with their corresponding interval notation:
1. \( x \leq 7.8 \) matches \( (-\infty, 7.8] \)
- This means \( x \) can take any value less than or equal to 7.8, including 7.8 itself.
2. \( x < 7.8 \) matches \( (-\infty, 7.8) \)
- This indicates that \( x \) can take any value less than 7.8, but not 7.8 itself.
3. \( x > 7.8 \) matches \( (7.8, \infty) \)
- This means \( x \) can take any value greater than 7.8, but not 7.8 itself.
4. \( x \geq 7.8 \) matches \( [7.8, \infty) \)
- This indicates that \( x \) can take any value greater than or equal to 7.8, including 7.8 itself.
So the final pairings are:
- \( x \leq 7.8 \) ↔ \( (-\infty, 7.8] \)
- \( x < 7.8 \) ↔ \( (-\infty, 7.8) \)
- \( x > 7.8 \) ↔ \( (7.8, \infty) \)
- \( x \geq 7.8 \) ↔ \( [7.8, \infty) \)
These matches are correct.
1. \( x \leq 7.8 \) matches \( (-\infty, 7.8] \)
- This means \( x \) can take any value less than or equal to 7.8, including 7.8 itself.
2. \( x < 7.8 \) matches \( (-\infty, 7.8) \)
- This indicates that \( x \) can take any value less than 7.8, but not 7.8 itself.
3. \( x > 7.8 \) matches \( (7.8, \infty) \)
- This means \( x \) can take any value greater than 7.8, but not 7.8 itself.
4. \( x \geq 7.8 \) matches \( [7.8, \infty) \)
- This indicates that \( x \) can take any value greater than or equal to 7.8, including 7.8 itself.
So the final pairings are:
- \( x \leq 7.8 \) ↔ \( (-\infty, 7.8] \)
- \( x < 7.8 \) ↔ \( (-\infty, 7.8) \)
- \( x > 7.8 \) ↔ \( (7.8, \infty) \)
- \( x \geq 7.8 \) ↔ \( [7.8, \infty) \)
These matches are correct.
Answered by
GPT-4o mini
To solve the inequality
\[
\frac{4}{9}x - 10 > \frac{x}{3} - 12,
\]
let's first eliminate the fractions by finding a common denominator. The least common multiple of 9 and 3 is 9. We can multiply every term by 9 to eliminate the fractions:
\[
9 \left(\frac{4}{9}x\right) - 9(10) > 9 \left(\frac{x}{3}\right) - 9(12).
\]
This simplifies to:
\[
4x - 90 > 3x - 108.
\]
Next, isolate \(x\) on one side. We can do this by subtracting \(3x\) from both sides:
\[
4x - 3x - 90 > -108,
\]
which simplifies to:
\[
x - 90 > -108.
\]
Now, add 90 to both sides:
\[
x > -18.
\]
This means the solution set for the inequality is:
\[
(-18, \infty).
\]
### Graphing on the Number Line
On a number line that ranges from -20 to 0, the point where \(x = -18\) is the starting point of the solution. Since the solution is \(x > -18\), you will represent this on the number line with an open circle (to indicate that -18 is not included in the solution) at -18, and shade or indicate the area to the right (towards positive values).
So, you should drag the point to just above -18 on the number line, indicating the beginning of the solution set, and indicate that the solution extends to the right.
\[
\frac{4}{9}x - 10 > \frac{x}{3} - 12,
\]
let's first eliminate the fractions by finding a common denominator. The least common multiple of 9 and 3 is 9. We can multiply every term by 9 to eliminate the fractions:
\[
9 \left(\frac{4}{9}x\right) - 9(10) > 9 \left(\frac{x}{3}\right) - 9(12).
\]
This simplifies to:
\[
4x - 90 > 3x - 108.
\]
Next, isolate \(x\) on one side. We can do this by subtracting \(3x\) from both sides:
\[
4x - 3x - 90 > -108,
\]
which simplifies to:
\[
x - 90 > -108.
\]
Now, add 90 to both sides:
\[
x > -18.
\]
This means the solution set for the inequality is:
\[
(-18, \infty).
\]
### Graphing on the Number Line
On a number line that ranges from -20 to 0, the point where \(x = -18\) is the starting point of the solution. Since the solution is \(x > -18\), you will represent this on the number line with an open circle (to indicate that -18 is not included in the solution) at -18, and shade or indicate the area to the right (towards positive values).
So, you should drag the point to just above -18 on the number line, indicating the beginning of the solution set, and indicate that the solution extends to the right.
Answered by
GPT-4o mini
Let's solve the inequality:
\[
x - 10 > -12
\]
To isolate \( x \), add 10 to both sides:
\[
x > -12 + 10
\]
This simplifies to:
\[
x > -2
\]
The solution to this inequality is \( (-2, \infty) \), meaning \( x \) can take any value greater than -2.
Now, looking at your options:
1. \( (–∞, –18) \)
2. \( (–18, ∞) \)
3. \( (–∞, 18) \)
4. \( (18, ∞) \)
None of these options correspond directly to \( (-2, \infty) \).
If you mean to determine the range that is indicated by your question, the correct solution based on the given inequality is \( (-2, \infty) \), which is not explicitly listed in your options. If there is a misunderstanding regarding the options presented, please double-check if any of the numbers need adjustment.
If the question was meant to be \( x - 10 > -12 \) and no transcription errors occurred, please correct the options accordingly for the specified solution.
\[
x - 10 > -12
\]
To isolate \( x \), add 10 to both sides:
\[
x > -12 + 10
\]
This simplifies to:
\[
x > -2
\]
The solution to this inequality is \( (-2, \infty) \), meaning \( x \) can take any value greater than -2.
Now, looking at your options:
1. \( (–∞, –18) \)
2. \( (–18, ∞) \)
3. \( (–∞, 18) \)
4. \( (18, ∞) \)
None of these options correspond directly to \( (-2, \infty) \).
If you mean to determine the range that is indicated by your question, the correct solution based on the given inequality is \( (-2, \infty) \), which is not explicitly listed in your options. If there is a misunderstanding regarding the options presented, please double-check if any of the numbers need adjustment.
If the question was meant to be \( x - 10 > -12 \) and no transcription errors occurred, please correct the options accordingly for the specified solution.
Answered by
GPT-4o mini
Let's solve the inequality
\[
\frac{4}{9}x - 10 > \frac{x}{3} - 12.
\]
1. First, we eliminate the fractions by finding a common denominator. The least common multiple of 9 and 3 is 9. We multiply every term in the inequality by 9:
\[
9 \left(\frac{4}{9}x\right) - 9(10) > 9 \left(\frac{x}{3}\right) - 9(12).
\]
This simplifies to:
\[
4x - 90 > 3x - 108.
\]
2. Next, isolate \(x\) by subtracting \(3x\) from both sides:
\[
4x - 3x - 90 > -108.
\]
This gives us:
\[
x - 90 > -108.
\]
3. Now, add 90 to both sides:
\[
x > -108 + 90.
\]
This simplifies to:
\[
x > -18.
\]
The solution to the inequality is:
\[
(-18, \infty).
\]
Now, looking at your options:
1. \( (–∞, –18) \)
2. \( (–18, ∞) \)
3. \( (–∞, 18) \)
4. \( (18, ∞) \)
The correct choice that corresponds to the solution \(x > -18\) is:
**(–18, ∞)**.
\[
\frac{4}{9}x - 10 > \frac{x}{3} - 12.
\]
1. First, we eliminate the fractions by finding a common denominator. The least common multiple of 9 and 3 is 9. We multiply every term in the inequality by 9:
\[
9 \left(\frac{4}{9}x\right) - 9(10) > 9 \left(\frac{x}{3}\right) - 9(12).
\]
This simplifies to:
\[
4x - 90 > 3x - 108.
\]
2. Next, isolate \(x\) by subtracting \(3x\) from both sides:
\[
4x - 3x - 90 > -108.
\]
This gives us:
\[
x - 90 > -108.
\]
3. Now, add 90 to both sides:
\[
x > -108 + 90.
\]
This simplifies to:
\[
x > -18.
\]
The solution to the inequality is:
\[
(-18, \infty).
\]
Now, looking at your options:
1. \( (–∞, –18) \)
2. \( (–18, ∞) \)
3. \( (–∞, 18) \)
4. \( (18, ∞) \)
The correct choice that corresponds to the solution \(x > -18\) is:
**(–18, ∞)**.
Answered by
GPT-4o mini
To determine how many items the company needs to sell to make a profit, we need to set up an inequality where the revenue is greater than the operating costs.
The revenue is given by:
\[
R = 7.5x - 100
\]
And the operating costs are given by:
\[
C = 79.86 + 5.8x
\]
To find the number of items that need to be sold to make a profit, we set up the inequality:
\[
R > C
\]
Substituting the expressions for revenue and costs into the inequality:
\[
7.5x - 100 > 79.86 + 5.8x
\]
Now, let's solve this inequality step by step:
1. Subtract \(5.8x\) from both sides:
\[
7.5x - 5.8x - 100 > 79.86
\]
This simplifies to:
\[
1.7x - 100 > 79.86
\]
2. Next, add 100 to both sides:
\[
1.7x > 79.86 + 100
\]
This simplifies to:
\[
1.7x > 179.86
\]
3. Now, divide both sides by 1.7:
\[
x > \frac{179.86}{1.7}
\]
Calculating the right-hand side:
\[
x > 105.2
\]
Since \(x\) represents the number of items sold, we round up to the next whole number (as you can't sell a fraction of an item):
The solution to the inequality is:
\[
x \geq 106
\]
Therefore, the company must sell at least **106 items** to make a profit.
To summarize:
- The inequality that will determine the number of items that need to be sold to make a profit is:
\[ 7.5x - 100 > 79.86 + 5.8x \]
- The solution to the inequality is:
\[ x > 105.2 \] (or \(x \geq 106\))
- The company must sell at least **106 items** to make a profit.
The revenue is given by:
\[
R = 7.5x - 100
\]
And the operating costs are given by:
\[
C = 79.86 + 5.8x
\]
To find the number of items that need to be sold to make a profit, we set up the inequality:
\[
R > C
\]
Substituting the expressions for revenue and costs into the inequality:
\[
7.5x - 100 > 79.86 + 5.8x
\]
Now, let's solve this inequality step by step:
1. Subtract \(5.8x\) from both sides:
\[
7.5x - 5.8x - 100 > 79.86
\]
This simplifies to:
\[
1.7x - 100 > 79.86
\]
2. Next, add 100 to both sides:
\[
1.7x > 79.86 + 100
\]
This simplifies to:
\[
1.7x > 179.86
\]
3. Now, divide both sides by 1.7:
\[
x > \frac{179.86}{1.7}
\]
Calculating the right-hand side:
\[
x > 105.2
\]
Since \(x\) represents the number of items sold, we round up to the next whole number (as you can't sell a fraction of an item):
The solution to the inequality is:
\[
x \geq 106
\]
Therefore, the company must sell at least **106 items** to make a profit.
To summarize:
- The inequality that will determine the number of items that need to be sold to make a profit is:
\[ 7.5x - 100 > 79.86 + 5.8x \]
- The solution to the inequality is:
\[ x > 105.2 \] (or \(x \geq 106\))
- The company must sell at least **106 items** to make a profit.
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