• A. Divide both sides of the inequality by 6 to simplify: x/6 - 5/6 <6
• B. Divide both sides of the inequality by x to simplify: 1-5/x <6
C. Multiply both sides of the inequality by x to simplify: x-5 < 6
•
D. Multiply both sides of the inequality by 6 to simplify: 6x-30 < x
11 answers
To solve the inequality \( x - 5 < \frac{x}{6} \), the correct first step to simplify the inequality involves eliminating the fraction.
The most straightforward approach is to multiply both sides by 6, which is option D.
So the first step is: D. Multiply both sides of the inequality by 6 to simplify: \( 6(x - 5) < x \).
This step will eliminate the fraction on the right side and help to simplify the inequality.
To solve the system of equations:
We can use the substitution or elimination method. Here, I'll use substitution.
From Equation 1, solve for \( x \): \[ x = 15 - 3y \]
Now substitute this expression for \( x \) into Equation 2: \[ 3(15 - 3y) + 2y = 10 \]
Distributing the 3: \[ 45 - 9y + 2y = 10 \]
Combine like terms: \[ 45 - 7y = 10 \]
Now, isolate \( y \): \[ -7y = 10 - 45 \] \[ -7y = -35 \]
Divide both sides by -7: \[ y = 5 \]
Thus, the value of \( y \) for the solution to the system is: \[ y = 5 \]
To determine which equation results in a system of equations with exactly one solution, we need to analyze the slope and y-intercept of the given linear equation and compare it with the options provided.
A system of equations has exactly one solution if the lines represented by the equations intersect at exactly one point, which occurs when the lines are neither parallel (same slope, different intercept) nor the same line (same slope and intercept).
Let’s analyze the given options to find their slopes.
\(A. -2x + 6y = -12\)
Rearranging to slope-intercept form (y = mx + b):
\[
6y = 2x - 12 \implies y = \frac{1}{3}x - 2
\]
(Slope = \(\frac{1}{3}\))
\(B. 3x - 9y = 9\)
Rearranging:
\[
-9y = -3x + 9 \implies y = \frac{1}{3}x - 1
\]
(Slope = \(\frac{1}{3}\))
\(C. x - 3y = 6\)
Rearranging:
\[
-3y = -x + 6 \implies y = \frac{1}{3}x + 2
\]
(Slope = \(\frac{1}{3}\))
\(D. 9x - 3y = 6\)
Rearranging:
\[
-3y = -9x + 6 \implies y = 3x - 2
\]
(Slope = \(3\))
Options A, B, and C have the same slope (\(\frac{1}{3}\)), making them parallel to the original line and resulting in no solution (as they will not intersect). However, option D has a different slope (3), meaning it will intersect with the original line and create exactly one solution.
Thus, the correct answer is:
D. \(9x - 3y = 6\)
To classify Joe's claim regarding the equation \( y = 2x + b \):
The equation \( y = mx + b \) is the slope-intercept form of a linear equation, where:
In Joe's equation, the slope \( m \) is \( 2 \) (a constant) and \( b \) is any real number that indicates the y-intercept.
Since linear equations in this form are valid for any value of \( b \), they will always represent straight lines.
Therefore, the BEST classification of Joe's claim is:
B. Joe's claim is true for all values of b.
Let \( q \) represent the number of quarters and \( d \) represent the number of dimes. We know from the problem:
The total number of coins is 124: \[ q + d = 124 \quad \text{(Equation 1)} \]
The total value in dollars is $25, which can be expressed in terms of cents as 2500 cents. Quarters are worth 25 cents each and dimes are worth 10 cents each: \[ 25q + 10d = 2500 \quad \text{(Equation 2)} \]
Now we can solve this system of equations.
Step 1: Solve Equation 1 for \( d \): \[ d = 124 - q \]
Step 2: Substitute \( d \) in Equation 2: \[ 25q + 10(124 - q) = 2500 \]
Step 3: Expand and simplify: \[ 25q + 1240 - 10q = 2500 \] \[ 15q + 1240 = 2500 \]
Step 4: Isolate \( q \): \[ 15q = 2500 - 1240 \] \[ 15q = 1260 \] \[ q = \frac{1260}{15} \] \[ q = 84 \]
So, Xavior had 84 quarters.
However, the answer choices provided are:
A. 40
B. 62
C. 841
D. 100
None of these choices match our calculation of 84 quarters. It seems there might be an inconsistency, or a miscommunication regarding the options.
If you'd like to double-check or if there's more information or an updated list of choices, please let me know!
To solve for the price that the miniature golf course charges for adults, we can set up a system of equations based on the information provided.
Let:
From the information provided, we have two scenarios:
On Saturday: \[ 50x + 50y = 800 \quad \text{(Equation 1)} \]
On Sunday: \[ 65x + 75y = 1100 \quad \text{(Equation 2)} \]
Step 1: Simplify Equation 1
Divide the entire equation by \( 50 \): \[ x + y = 16 \quad \text{(Equation 3)} \]
Step 2: Simplify Equation 2
Divide the entire equation by \( 5 \): \[ 13x + 15y = 220 \quad \text{(Equation 4)} \]
Step 3: Solve the System of Equations
Now we can use Equation 3 to express \( y \) in terms of \( x \): \[ y = 16 - x \]
Step 4: Substitute \( y \) into Equation 4: \[ 13x + 15(16 - x) = 220 \] Expanding this: \[ 13x + 240 - 15x = 220 \] Combine like terms: \[ -2x + 240 = 220 \] Isolate \( x \): \[ -2x = 220 - 240 \] \[ -2x = -20 \] \[ x = 10 \]
So, the price charged for adults is $10.
Thus, the answer is:
C. $10
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