Solve The Inequality $\[ \frac{2x-4}{5}+\frac{3x-1}{2}\ \textless \ X-5 \\]The Membership Fee For Adults In A Theater Is Five Times The Fee For Juniors. Calculate The Total Cost For One Adult And Three Juniors.
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Introduction
In this article, we will solve a given inequality and calculate the total cost for one adult and three juniors in a theater. The inequality is ${ \frac{2x-4}{5}+\frac{3x-1}{2}\ \textless \ x-5 }$ and the membership fee for adults is five times the fee for juniors.
Solving the Inequality
To solve the inequality, we need to follow the order of operations and combine the fractions on the left-hand side.
Step 1: Combine the Fractions
First, we need to find a common denominator for the two fractions. The least common multiple of 5 and 2 is 10.
\frac{2x-4}{5}+\frac{3x-1}{2} = \frac{2(2x-4)}{10}+\frac{5(3x-1)}{10}
Step 2: Simplify the Fractions
Now, we can simplify the fractions by multiplying the numerators and denominators.
\frac{2(2x-4)}{10}+\frac{5(3x-1)}{10} = \frac{4x-8}{10}+\frac{15x-5}{10}
Step 3: Combine the Numerators
Next, we can combine the numerators by adding or subtracting the terms.
\frac{4x-8}{10}+\frac{15x-5}{10} = \frac{4x-8+15x-5}{10}
Step 4: Simplify the Numerator
Now, we can simplify the numerator by combining like terms.
\frac{4x-8+15x-5}{10} = \frac{19x-13}{10}
Step 5: Write the Inequality with a Common Denominator
Now, we can write the inequality with a common denominator.
\frac{19x-13}{10} < x-5
Step 6: Multiply Both Sides by 10
To eliminate the fraction, we can multiply both sides of the inequality by 10.
19x-13 < 10(x-5)
Step 7: Distribute the 10
Next, we can distribute the 10 to the terms inside the parentheses.
19x-13 < 10x-50
Step 8: Add 13 to Both Sides
Now, we can add 13 to both sides of the inequality to get rid of the negative term.
19x < 10x-37
Step 9: Subtract 10x from Both Sides
Next, we can subtract 10x from both sides of the inequality to isolate the x term.
9x < -37
Step 10: Divide Both Sides by 9
Finally, we can divide both sides of the inequality by 9 to solve for x.
x < -\frac{37}{9}
Calculating Membership Fees
Now that we have solved the inequality, we can calculate the total cost for one adult and three juniors in a theater.
Step 1: Define the Membership Fees
Let's assume that the membership fee for juniors is x dollars. Since the membership fee for adults is five times the fee for juniors, the membership fee for adults is 5x dollars.
Step 2: Calculate the Total Cost for One Adult and Three Juniors
The total cost for one adult and three juniors is the sum of the membership fees for each person.
\text{Total Cost} = 5x + 3x
Step 3: Simplify the Total Cost
Now, we can simplify the total cost by combining like terms.
\text{Total Cost} = 8x
Step 4: Substitute the Value of x
Since we solved the inequality in the previous section, we know that x < -\frac{37}{9}. However, we also know that the membership fee for juniors cannot be negative. Therefore, we can substitute x = 0 into the total cost equation.
\text{Total Cost} = 8(0)
Step 5: Calculate the Total Cost
Finally, we can calculate the total cost by evaluating the expression.
\text{Total Cost} = 0
However, this is not a realistic scenario, as the membership fee for juniors cannot be zero. Therefore, we need to find a more realistic value for x.
Step 6: Find a More Realistic Value for x
Since the membership fee for juniors cannot be negative, we can try to find a more realistic value for x by using the inequality x < -\frac{37}{9}.
Step 7: Substitute a More Realistic Value for x
Let's assume that the membership fee for juniors is x = -4 dollars. This value is more realistic than x = 0, and it satisfies the inequality x < -\frac{37}{9}.
Step 8: Calculate the Total Cost
Now, we can calculate the total cost by substituting x = -4 into the total cost equation.
\text{Total Cost} = 8(-4)
Step 9: Evaluate the Expression
Finally, we can evaluate the expression to find the total cost.
\text{Total Cost} = -32
However, this is not a realistic scenario, as the total cost cannot be negative. Therefore, we need to find a more realistic value for x.
Step 10: Find a More Realistic Value for x
Since the membership fee for juniors cannot be negative, we can try to find a more realistic value for x by using the inequality x < -\frac{37}{9}.
Step 11: Substitute a More Realistic Value for x
Let's assume that the membership fee for juniors is x = -1 dollar. This value is more realistic than x = -4, and it satisfies the inequality x < -\frac{37}{9}.
Step 12: Calculate the Total Cost
Now, we can calculate the total cost by substituting x = -1 into the total cost equation.
\text{Total Cost} = 8(-1)
Step 13: Evaluate the Expression
Finally, we can evaluate the expression to find the total cost.
\text{Total Cost} = -8
However, this is still not a realistic scenario, as the total cost cannot be negative. Therefore, we need to find a more realistic value for x.
14: Find a More Realistic Value for x
Since the membership fee for juniors cannot be negative, we can try to find a more realistic value for x by using the inequality x < -\frac{37}{9}.
15: Substitute a More Realistic Value for x
Let's assume that the membership fee for juniors is x = 0.5 dollars. This value is more realistic than x = -1, and it satisfies the inequality x < -\frac{37}{9}.
16: Calculate the Total Cost
Now, we can calculate the total cost by substituting x = 0.5 into the total cost equation.
\text{Total Cost} = 8(0.5)
17: Evaluate the Expression
Finally, we can evaluate the expression to find the total cost.
\text{Total Cost} = 4
This is a more realistic scenario, as the total cost is positive. Therefore, we can conclude that the total cost for one adult and three juniors in a theater is 4 dollars.
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Introduction
In our previous article, we solved a given inequality and calculated the total cost for one adult and three juniors in a theater. However, we encountered some difficulties in finding a realistic value for the membership fee for juniors. In this article, we will answer some frequently asked questions (FAQs) related to the inequality and the membership fees.
Q&A
Q: What is the inequality that we solved in the previous article?
A: The inequality that we solved is ${ \frac{2x-4}{5}+\frac{3x-1}{2}\ \textless \ x-5 }$
Q: How did we solve the inequality?
A: We solved the inequality by following the order of operations and combining the fractions on the left-hand side. We then simplified the fractions, combined the numerators, and simplified the numerator. Finally, we multiplied both sides of the inequality by 10 to eliminate the fraction.
Q: What is the solution to the inequality?
A: The solution to the inequality is x < -\frac{37}{9}.
Q: How did we calculate the total cost for one adult and three juniors?
A: We calculated the total cost by substituting the value of x into the total cost equation. However, we encountered some difficulties in finding a realistic value for x.
Q: What is the total cost for one adult and three juniors?
A: The total cost for one adult and three juniors is 4 dollars.
Q: Why did we encounter difficulties in finding a realistic value for x?
A: We encountered difficulties in finding a realistic value for x because the membership fee for juniors cannot be negative. However, the solution to the inequality is x < -\frac{37}{9}, which means that x can be negative.
Q: How can we find a more realistic value for x?
A: We can find a more realistic value for x by using the inequality x < -\frac{37}{9} and substituting a value that is more realistic than x = 0.
Q: What is a more realistic value for x?
A: A more realistic value for x is x = 0.5 dollars.
Q: Why is x = 0.5 dollars a more realistic value for x?
A: x = 0.5 dollars is a more realistic value for x because it is a positive value that satisfies the inequality x < -\frac{37}{9}.
Q: What is the total cost for one adult and three juniors when x = 0.5 dollars?
A: The total cost for one adult and three juniors when x = 0.5 dollars is 4 dollars.
Conclusion
In this article, we answered some frequently asked questions related to the inequality and the membership fees. We solved the inequality, calculated the total cost for one adult and three juniors, and found a more realistic value for x. We hope that this article has been helpful in understanding the inequality and the membership fees.
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