Considering the one-time deposit and monthly rent, the equation c = 20m + 25 can be used to represent the total cost of a gym membership for m months. What is the amount of the sign-up fee?
31 answers
The sign-up fee is represented by the number 25 in the equation.
Which is an equivalent form of the equation 2x - 4a = 10?
An equivalent form of the equation 2x - 4a = 10 can be obtained by dividing all the terms by 2: x - 2a = 5.
So the answer is x = 5 - 2a?
No, the answer is x - 2a = 5.
That's not an option. What equation is equivalent to 2x - 4a = 10?
x = 5/4a
x = 5 + 2a
x = 10 - 4a
x = 5 - 2a
x = 5/4a
x = 5 + 2a
x = 10 - 4a
x = 5 - 2a
Apologies for the confusion. The correct equivalent equation to 2x - 4a = 10 is:
x = 5 + 2a
x = 5 + 2a
Name the property of real numbers illustrated by the equation.
pi x 3 = 3 x pi
pi x 3 = 3 x pi
The property illustrated by the equation "pi x 3 = 3 x pi" is the commutative property of multiplication.
Which algebraic expression models the given word phrase?
6 times the sum of a and b
6 times the sum of a and b
The algebraic expression that models the word phrase "6 times the sum of a and b" is 6(a + b).
John has $305, and he is spending $3 each day. Which algebraic expression describes this situation, where d represents the number of days?
3d + 305
3 - 305d
305d + 3
305 - 3d
3d + 305
3 - 305d
305d + 3
305 - 3d
The algebraic expression that describes the situation is: 3d + 305.
Solve the equation or formula for the indicated variable.
S = 3r^4t, for t
S = 3r^4t, for t
To solve the equation for t, we need to isolate t on one side of the equation. Here's the step-by-step process:
Step 1: Divide both sides of the equation by 3r^4:
S/(3r^4) = t
Therefore, the equation, S = 3r^4t, when solved for t, is t = S/(3r^4).
Step 1: Divide both sides of the equation by 3r^4:
S/(3r^4) = t
Therefore, the equation, S = 3r^4t, when solved for t, is t = S/(3r^4).
Which inequality represents the following sentence?
Seven fewer than a number is at least 14.
x - 7 < 14
7 - x ≥ 14
x + 7 ≤ 14
x - 7 ≥ 14
Seven fewer than a number is at least 14.
x - 7 < 14
7 - x ≥ 14
x + 7 ≤ 14
x - 7 ≥ 14
The inequality that represents the sentence "Seven fewer than a number is at least 14" is:
x - 7 ≥ 14
x - 7 ≥ 14
Solve the absolute value equation.
|4x - 1| = 1
|4x - 1| = 1
To solve the given absolute value equation |4x - 1| = 1, we can break it down into two separate equations:
1) 4x - 1 = 1
2) 4x - 1 = -1
Solving Equation 1:
4x - 1 = 1
Adding 1 to both sides:
4x = 2
Dividing by 4:
x = 1/2
Solving Equation 2:
4x - 1 = -1
Adding 1 to both sides:
4x = 0
Dividing by 4:
x = 0
So, the solution to the absolute value equation |4x - 1| = 1 is x = 1/2 and x = 0.
1) 4x - 1 = 1
2) 4x - 1 = -1
Solving Equation 1:
4x - 1 = 1
Adding 1 to both sides:
4x = 2
Dividing by 4:
x = 1/2
Solving Equation 2:
4x - 1 = -1
Adding 1 to both sides:
4x = 0
Dividing by 4:
x = 0
So, the solution to the absolute value equation |4x - 1| = 1 is x = 1/2 and x = 0.
Which represents the solution to the absolute value equation?
2|x + 2| - 1 = 5
2|x + 2| - 1 = 5
To solve the given absolute value equation 2|x + 2| - 1 = 5, we need to isolate the absolute value term and then solve for x using two cases.
Case 1: x + 2 ≥ 0
When x + 2 ≥ 0, the absolute value can be removed by keeping the expression inside the absolute value unchanged. So, we have:
2(x + 2) - 1 = 5
2x + 4 - 1 = 5
2x + 3 = 5
2x = 2
x = 1
Case 2: x + 2 < 0
When x + 2 < 0, the absolute value becomes -1 times the expression inside. So, we have:
2(-1)(x + 2) - 1 = 5
-2x - 4 - 1 = 5
-2x - 5 = 5
-2x = 10
x = -5
Therefore, the solutions to the absolute value equation 2|x + 2| - 1 = 5 are x = 1 and x = -5.
Case 1: x + 2 ≥ 0
When x + 2 ≥ 0, the absolute value can be removed by keeping the expression inside the absolute value unchanged. So, we have:
2(x + 2) - 1 = 5
2x + 4 - 1 = 5
2x + 3 = 5
2x = 2
x = 1
Case 2: x + 2 < 0
When x + 2 < 0, the absolute value becomes -1 times the expression inside. So, we have:
2(-1)(x + 2) - 1 = 5
-2x - 4 - 1 = 5
-2x - 5 = 5
-2x = 10
x = -5
Therefore, the solutions to the absolute value equation 2|x + 2| - 1 = 5 are x = 1 and x = -5.
Solve the inequality. Graph the solution.
|2x + 7| ≤ 27
|2x + 7| ≤ 27
To solve the inequality |2x + 7| ≤ 27, we need to consider both possibilities when the expression inside the absolute value is positive or negative.
Case 1: 2x + 7 ≥ 0
When 2x + 7 ≥ 0, the absolute value can be removed, so we have:
2x + 7 ≤ 27
Subtracting 7 from both sides:
2x ≤ 20
Dividing both sides by 2:
x ≤ 10
Case 2: 2x + 7 < 0
When 2x + 7 < 0, the absolute value becomes -1 times the expression inside:
-(2x + 7) ≤ 27
-2x - 7 ≤ 27
Subtracting 7 from both sides:
-2x ≤ 34
Dividing both sides by -2 (and reversing the inequality):
x ≥ -17
Combining the two cases, the solution to the inequality |2x + 7| ≤ 27 is -17 ≤ x ≤ 10.
Now, let's graph the solution on a number line:
------------------●-------------●------------------
-17 10
The solid dots represent the values -17 and 10, which are included in the solution. The line segment between them represents the values of x that satisfy the inequality.
Case 1: 2x + 7 ≥ 0
When 2x + 7 ≥ 0, the absolute value can be removed, so we have:
2x + 7 ≤ 27
Subtracting 7 from both sides:
2x ≤ 20
Dividing both sides by 2:
x ≤ 10
Case 2: 2x + 7 < 0
When 2x + 7 < 0, the absolute value becomes -1 times the expression inside:
-(2x + 7) ≤ 27
-2x - 7 ≤ 27
Subtracting 7 from both sides:
-2x ≤ 34
Dividing both sides by -2 (and reversing the inequality):
x ≥ -17
Combining the two cases, the solution to the inequality |2x + 7| ≤ 27 is -17 ≤ x ≤ 10.
Now, let's graph the solution on a number line:
------------------●-------------●------------------
-17 10
The solid dots represent the values -17 and 10, which are included in the solution. The line segment between them represents the values of x that satisfy the inequality.
Solve the inequality. Graph the solution.
|x - 1| > 7
|x - 1| > 7
To solve the inequality |x - 1| > 7, we need to consider both possibilities when the expression inside the absolute value is positive or negative.
Case 1: x - 1 ≥ 0
When x - 1 ≥ 0, the absolute value can be removed, so we have:
x - 1 > 7
Adding 1 to both sides:
x > 8
Case 2: x - 1 < 0
When x - 1 < 0, the absolute value becomes -1 times the expression inside:
-(x - 1) > 7
-x + 1 > 7
Subtracting 1 from both sides:
-x > 6
Dividing both sides by -1 (and reversing the inequality):
x < -6
Combining the two cases, the solution to the inequality |x - 1| > 7 is x < -6 or x > 8.
Now, let's graph the solution on a number line:
--------------------------●------●------------------
-6 8
The open points represent the values -6 and 8, which are not included in the solution. The line segments on either side of the points represent the values of x that satisfy the inequality.
Case 1: x - 1 ≥ 0
When x - 1 ≥ 0, the absolute value can be removed, so we have:
x - 1 > 7
Adding 1 to both sides:
x > 8
Case 2: x - 1 < 0
When x - 1 < 0, the absolute value becomes -1 times the expression inside:
-(x - 1) > 7
-x + 1 > 7
Subtracting 1 from both sides:
-x > 6
Dividing both sides by -1 (and reversing the inequality):
x < -6
Combining the two cases, the solution to the inequality |x - 1| > 7 is x < -6 or x > 8.
Now, let's graph the solution on a number line:
--------------------------●------●------------------
-6 8
The open points represent the values -6 and 8, which are not included in the solution. The line segments on either side of the points represent the values of x that satisfy the inequality.
Write an inequality that models a real-world situation. Describe your situation and what the variable x represents.
In a real-world situation, let's say that x represents the number of hours a student studies for a test. And let's say that the situation involves earning a certain grade on the test.
The inequality that models this situation could be:
2x + 5 ≥ 90
In this inequality, x represents the number of hours the student studies, and we are stating that the sum of 2 times the number of hours studied plus 5 must be greater than or equal to 90. This is indicating that the student needs to earn a grade of at least 90 on the test by studying for a certain number of hours.
The inequality that models this situation could be:
2x + 5 ≥ 90
In this inequality, x represents the number of hours the student studies, and we are stating that the sum of 2 times the number of hours studied plus 5 must be greater than or equal to 90. This is indicating that the student needs to earn a grade of at least 90 on the test by studying for a certain number of hours.
Evaluate the expression for the given variable. Please show your steps.
|-4 - 8| + |-1 - b^2| + 2b; with b =-3
|-4 - 8| + |-1 - b^2| + 2b; with b =-3
To evaluate the expression |-4 - 8| + |-1 - b^2| + 2b with b = -3, we substitute -3 for b into the expression and perform the necessary calculations.
Starting with the first absolute value:
|-4 - 8| + |-1 - b^2| + 2b
= |-12| + |-1 - (-3)^2| + 2(-3)
|-12| simplifies to 12:
= 12 + |-1 - (-3)^2| + 2(-3)
(-3)^2 simplifies to 9:
= 12 + |-1 - 9| + 2(-3)
|-1 - 9| simplifies to 10:
= 12 + |10| + 2(-3)
|10| simplifies to 10:
= 12 + 10 + 2(-3)
2(-3) simplifies to -6:
= 12 + 10 - 6
Performing addition:
= 22 - 6
Finally, subtracting:
= 16
Therefore, when b = -3, the expression |-4 - 8| + |-1 - b^2| + 2b evaluates to 16.
Starting with the first absolute value:
|-4 - 8| + |-1 - b^2| + 2b
= |-12| + |-1 - (-3)^2| + 2(-3)
|-12| simplifies to 12:
= 12 + |-1 - (-3)^2| + 2(-3)
(-3)^2 simplifies to 9:
= 12 + |-1 - 9| + 2(-3)
|-1 - 9| simplifies to 10:
= 12 + |10| + 2(-3)
|10| simplifies to 10:
= 12 + 10 + 2(-3)
2(-3) simplifies to -6:
= 12 + 10 - 6
Performing addition:
= 22 - 6
Finally, subtracting:
= 16
Therefore, when b = -3, the expression |-4 - 8| + |-1 - b^2| + 2b evaluates to 16.
Solve the equation. Please show your steps and check for extraneous solutions.
4|9 - 5x| = 4x + 12
4|9 - 5x| = 4x + 12
To solve the equation 4|9 - 5x| = 4x + 12, we need to consider two cases based on the expression inside the absolute value.
Case 1: 9 - 5x ≥ 0
When 9 - 5x ≥ 0, the absolute value can be removed, so we have:
4(9 - 5x) = 4x + 12
Distribute on the left side:
36 - 20x = 4x + 12
Add 20x to both sides:
36 = 24x + 12
Subtract 12 from both sides:
24 = 24x
Divide both sides by 24:
x = 1
Case 2: 9 - 5x < 0
When 9 - 5x < 0, the absolute value becomes -1 times the expression inside:
-4(9 - 5x) = 4x + 12
Distribute on the left side:
-36 + 20x = 4x + 12
Subtract 4x and 12 from both sides:
20x - 4x = 12 + 36
16x = 48
Divide both sides by 16:
x = 3
Therefore, the solutions to the equation 4|9 - 5x| = 4x + 12 are x = 1 and x = 3.
Now, let's check for extraneous solutions by substituting the solutions back into the original equation.
For x = 1:
Left side: 4|9 - 5(1)| = 4|9 - 5| = 4|4| = 16
Right side: 4(1) + 12 = 4 + 12 = 16
For x = 3:
Left side: 4|9 - 5(3)| = 4|9 - 15| = 4|-6| = 24
Right side: 4(3) + 12 = 12 + 12 = 24
Since the left side and right side are equal for both solutions, there are no extraneous solutions.
Therefore, the solutions to the equation 4|9 - 5x| = 4x + 12 are x = 1 and x = 3.
Case 1: 9 - 5x ≥ 0
When 9 - 5x ≥ 0, the absolute value can be removed, so we have:
4(9 - 5x) = 4x + 12
Distribute on the left side:
36 - 20x = 4x + 12
Add 20x to both sides:
36 = 24x + 12
Subtract 12 from both sides:
24 = 24x
Divide both sides by 24:
x = 1
Case 2: 9 - 5x < 0
When 9 - 5x < 0, the absolute value becomes -1 times the expression inside:
-4(9 - 5x) = 4x + 12
Distribute on the left side:
-36 + 20x = 4x + 12
Subtract 4x and 12 from both sides:
20x - 4x = 12 + 36
16x = 48
Divide both sides by 16:
x = 3
Therefore, the solutions to the equation 4|9 - 5x| = 4x + 12 are x = 1 and x = 3.
Now, let's check for extraneous solutions by substituting the solutions back into the original equation.
For x = 1:
Left side: 4|9 - 5(1)| = 4|9 - 5| = 4|4| = 16
Right side: 4(1) + 12 = 4 + 12 = 16
For x = 3:
Left side: 4|9 - 5(3)| = 4|9 - 15| = 4|-6| = 24
Right side: 4(3) + 12 = 12 + 12 = 24
Since the left side and right side are equal for both solutions, there are no extraneous solutions.
Therefore, the solutions to the equation 4|9 - 5x| = 4x + 12 are x = 1 and x = 3.
2 answers