A number less 7 is greater than -45. How can the solutions for this inequality be described? Is -40 a solution?
A) All numbers less than -38.; yes
B) All numbers greater than -38.; no
C) All numbers less than or equal to -38.; yes
D) All numbers greater than or equal to -38.; no
22)
Katie's savings account balance is $1625. During the summer she will be making weekly withdrawals of $125. She needs to keep a balance of at least $500 in her account. How many withdrawals can Katie make during the summer and still maintain a $500 balance? If the solution were graphed on a number line, which number(s) would not fit the "context" of the problem?
A) w ≤ 9; 0
B) w ≤ 11; 0
C) w ≤ 8; negative numbers
D) w ≤ 9; negative numbers
I dont know how to get these
n-7>-45
add 7 to each side
n is greater than -38
balance=1625-125w
500<=balance, so
500<=1625-125w add 125w to each side, subtract 500 from each side
125w<=1125
w<=10
For question 1, the inequality is "a number less 7 is greater than -45". We can represent this as x - 7 > -45. To solve this inequality, we add 7 to both sides: x > -45 + 7. Simplifying, we get x > -38. Therefore, the solutions to this inequality can be described as "All numbers greater than -38". Therefore, option B) "All numbers greater than -38; no" is the correct answer. -40 is not a solution since it is not greater than -38.
For question 2, Katie's savings account balance is $1625 and she will be making weekly withdrawals of $125. She needs to keep a balance of at least $500 in her account. To find out how many withdrawals she can make and still maintain a $500 balance, we can set up an inequality: 1625 - 125w ≥ 500, where w represents the number of withdrawals.
Let's solve this inequality to find the maximum value of w. Subtract 1625 from both sides: -125w ≥ 500 - 1625. Simplifying, we get -125w ≥ -1125. To isolate w, divide both sides by -125 (remember to flip the inequality sign when dividing by a negative number): w ≤ -1125 / -125. Simplifying further, w ≤ 9.
So, Katie can make a maximum of 9 withdrawals while still maintaining a $500 balance. Therefore, option A) "w ≤ 9; 0" is the correct answer. None of the other options fit the context of the problem since they either allow for a greater number of withdrawals or include negative numbers, which is not applicable in this scenario.
To solve these problems, we need to analyze the given conditions and determine the ranges of values that satisfy them.
1) A number less than 7 is greater than -45. Let's set up the inequality:
x < 7 > -45
To solve this inequality, we can simplify it by eliminating the double inequality sign. Since -45 is already less than 7, we can rewrite the inequality as:
x > -45
The solution for this inequality is "All numbers greater than -45." Now, let's check if -40 is a solution:
-40 > -45
Since -40 is indeed greater than -45, -40 is a solution.
So, the answer to this problem is:
C) All numbers less than or equal to -38.; yes
2) Katie has a savings account balance of $1625 and will be making weekly withdrawals of $125. She needs to maintain a balance of at least $500.
Let's set up the inequality to represent the balance after the withdrawals:
1625 - 125w ≥ 500
Solving this inequality, we can isolate the variable 'w':
-125w ≥ 500 - 1625
-125w ≥ -1125
w ≤ -1125/-125
w ≤ 9
The solution to this inequality is "w is less than or equal to 9." This means Katie can make a maximum of 9 withdrawals while still maintaining a $500 balance.
Now, let's analyze the answer choices to determine which number(s) would not fit the context of the problem:
A) w ≤ 9; 0: This answer indicates that Katie can make a maximum of 9 withdrawals, which is correct.
B) w ≤ 11; 0: This answer allows for more than 9 withdrawals, so it does not fit the problem's context.
C) w ≤ 8; negative numbers: This answer allows for fewer withdrawals and also includes negative numbers, which are not applicable in this situation.
D) w ≤ 9; negative numbers: This answer includes negative numbers, which are not applicable.
Therefore, the answer is B) w ≤ 11; 0, as it does not fit the "context" of the problem.
To solve the first inequality, "A number less 7 is greater than -45," we can follow these steps:
1. Write the inequality:
x - 7 > -45
2. Add 7 to both sides of the inequality to isolate the variable:
x - 7 + 7 > -45 + 7
x > -38
This means that any number greater than -38 is a solution to the inequality. Now, let's check if -40 is a solution:
Plugging in -40 into the inequality: -40 - 7 > -45
-47 > -45
Since the statement is true, -40 is indeed a solution to the inequality.
Therefore, the correct answer is:
A) All numbers less than -38.; yes
Now let's move on to the second question about Katie's savings account balance.
To find the number of withdrawals Katie can make during the summer while maintaining a $500 balance, we can use the following steps:
1. Calculate the total amount of money Katie will withdraw during the summer:
Amount withdrawn = Weekly withdrawal x Number of weeks
Amount withdrawn = $125 x Number of weeks
2. Set up an inequality to represent the balance she needs to maintain:
Account balance ≥ $500
3. Substitute the equation from step 1 into the inequality from step 2:
$1625 - (Weekly withdrawal x Number of weeks) ≥ $500
4. Solve the inequality for the maximum number of withdrawals:
$1625 - $500 ≥ Weekly withdrawal x Number of weeks
$1125 ≥ Weekly withdrawal x Number of weeks
5. Divide both sides of the inequality by the weekly withdrawal:
$1125 / $125 ≥ Number of weeks
9 ≥ Number of weeks
This means that Katie can make a maximum of 9 withdrawals during the summer while still maintaining a $500 balance.
Now let's see which answer choice represents the number of withdrawals and does not fit the problem's context:
The correct answer is:
B) w ≤ 11; 0
This answer represents that Katie can make up to 11 withdrawals, which exceeds the maximum allowed withdrawals of 9.