Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate, how many minutes will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:
Solution:
Meaning of the solution in words:
Let x be the number of minutes it takes Grandma to remove 30 weeds from the garden.
We know that she can remove weeds at a rate of 5 per minute.
So in x minutes, Grandma can remove 5x weeds from the garden.
Therefore, the equation can be written as 250 - 5x = 30.
Simplifying the equation, we get 220 - 5x = 0.
Moving the terms around, we get 5x = 220.
Dividing both sides by 5, we get x = 44.
Therefore, it will take Grandma 44 minutes to remove 30 weeds from the garden.
The Celsius to Fahrenheit formula: F=9/5C+32
Solve this formula for C. Check all answers that are true.
(4 points)
Responses
First step is to multiply by 9 on each side.
First step is to add 32 to both sides.
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to divide both sides by 5
Solution: 5/9F - 32 = C
To solve this equation for C, you would have to distribute at some point.
The correct steps to solve the formula F = 9/5C + 32 for C are as follows:
1. Subtract 32 from both sides: F - 32 = 9/5C
2. To isolate C, multiply both sides by 5/9 (or divide by 9/5): (F - 32) * (5/9) = C
So, the correct solution is C = (F - 32) * (5/9).
Checking all answers:
1. First step is to multiply by 9 on each side. - Not true, the first step is to subtract 32 from both sides.
2. First step is to add 32 to both sides. - Not true, the first step is to subtract 32 from both sides.
3. The next step from F - 32 = 9/5 C, could be to multiply both sides by 5. - Not true, the next step is to multiply both sides by 5/9.
4. The next step from F - 32 = 9/5 C, could be to divide both sides by 5. - True, this is the correct step.
The correct solution is C = (F - 32) * (5/9).
The Celsius to Fahrenheit formula: F=9/5C+32
Solve this formula for C. Check all answers that are true.
(4 points)
Responses
1 First step is to multiply by 9 on each side.
2 First step is to add 32 to both sides.
3 The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
4 The next step from F - 32 = 9/5 C, could be to divide both sides by 5
5 Solution: 5/9F - 32 = C
6 To solve this equation for C, you would have to distribute at some point.
To solve the Celsius to Fahrenheit formula F = (9/5)C + 32 for C, we need to perform the following steps:
1. First step is to subtract 32 from both sides: F - 32 = (9/5)C
2. The next step is to multiply both sides by 5/9 (or divide by 9/5): (F - 32) * (5/9) = C
So, the correct solution is C = (F - 32) * (5/9).
Now, let's check which statements are true:
1. First step is to multiply by 9 on each side. - False, the first step is to subtract 32 from both sides, not multiply by 9.
2. First step is to add 32 to both sides. - False, the first step is to subtract 32 from both sides, not add 32.
3. The next step from F - 32 = (9/5)C, could be to multiply both sides by 5. - False, the next step is to multiply both sides by 5/9, not 5.
4. The next step from F - 32 = (9/5)C, could be to divide both sides by 5. - False, the next step is to multiply both sides by 5/9, not divide by 5.
5. Solution: (5/9)F - 32 = C - True, this is the correct solution for C.
6. To solve this equation for C, you would have to distribute at some point. - False, there is no need to distribute in this equation.
Therefore, only Statement 5 is true: Solution: (5/9)F - 32 = C.
Izzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you and asks you if she did it correctly.
This is the work she completed:
Step 1: 3t+3−4t≥−5
Step 2: −t+3≥−5
Step 3: −t≥−2
Step 4: t≤2
Izzie's work is partially correct, but she made a mistake in Step 3. Let's go through the steps again:
Step 1: Distribute 3 to both terms inside the parentheses:
3t + 3 - 4t ≥ -5
Step 2: Combine like terms:
-t + 3 ≥ -5
Step 3: Subtract 3 from both sides:
-t ≥ -5 - 3
-t ≥ -8
At this point, we need to remember that when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign needs to be flipped. Since we are dividing both sides by -1 (or multiplying by -1), the inequality sign will change direction.
Step 4: Multiply both sides by -1 (or divide by -1) and flip the inequality sign:
t ≤ 8
So the correct solution is t ≤ 8, not t ≤ 2.
Therefore, Izzie made a mistake in Step 4. The correct inequality is t ≤ 8.
5n−3<3(n+1)
To solve the inequality 5n - 3 < 3(n + 1), we will use the distributive property and then simplify the expression:
Step 1: Distribute 3 to both terms inside the parentheses:
5n - 3 < 3n + 3
Step 2: Combine like terms on both sides of the inequality:
5n - 3n < 3 + 3
Step 3: Simplify both sides of the equation:
2n < 6
Step 4: Divide both sides by 2 (the coefficient of n) to isolate n:
n < 6/2
Step 5: Simplify the right side of the equation:
n < 3
Therefore, the solution to the inequality is n < 3.
Solve the equation and inequality for x.
Part A: Solve the equation for x showing all steps for credit: ax−5=18
Part B: Solve the inequality for x showing all steps for credit: ax−5>18
Part C: State the difference between an equation and an inequality regarding solving steps and solution
Part A:
To solve the equation ax - 5 = 18 for x, we will perform the following steps:
Step 1: Add 5 to both sides of the equation to isolate the variable term:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by a to solve for x:
ax/a = 23/a
x = 23/a
Therefore, the solution to the equation ax - 5 = 18 is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18 for x, we will perform the following steps:
Step 1: Add 5 to both sides of the inequality to isolate the variable term:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by a, considering that a can be positive or negative:
If a is positive: (ax)/a > 23/a
x > 23/a
If a is negative: (ax)/a < 23/a
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 depends on the sign of a. If a is positive, the solution is x > 23/a. If a is negative, the solution is x < 23/a.
Part C:
The difference between an equation and an inequality lies in the solution set and the steps required to find the solution:
1. Steps: Solving an equation involves performing various operations to find the value of the variable that makes both sides of the equation equal. On the other hand, solving an inequality involves finding the range of values for the variable that satisfies the given relationship (e.g., greater than, less than, greater than or equal to, less than or equal to).
2. Solution set: For equations, there is either one solution that satisfies the equation or no solution. For inequalities, there is often a range of values that satisfies the given relationship. The solution set for an inequality is usually represented by an inequality statement or an interval.
In summary, equations aim to find a specific value or values that satisfy the equation, whereas inequalities aim to find a range of values that satisfy the inequality statement.