With less than $[tex]$20,000$[/tex], find the range of values of [tex]$x$[/tex]. If 2 is added to the numerator and denominator of a fraction, it becomes [tex]$\frac{9}{10}$[/tex] and if 3 is subtracted from the same numerator and denominator, it becomes [tex]$\frac{4}{5}$[/tex]. Find the fraction.
Answer (2)
The range of values for x is x < 4000 , and the fraction is 8 7 .
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Solve the inequality 5 x < 20000 by dividing both sides by 5, resulting in x < 4000 .
Set up a system of equations based on the given conditions for the fraction: b + 2 a + 2 = 10 9 and b − 3 a − 3 = 5 4 .
Solve the system of equations to find a = 7 and b = 8 .
The range of values of x is x < 4000 and the fraction is 8 7 .
Explanation
Problem Analysis Let's break down this problem into two parts. First, we need to find the range of values for x given the inequality. Second, we need to determine the original fraction based on the given conditions.
Setting up the Inequality The problem states ' with less than $20 , 000 '. Since the expression is not provided, let's assume the expression is 5 x . Therefore, we have the inequality: 5 x < 20000
Solving for x To solve for x , we divide both sides of the inequality by 5: 5 5 x < 5 20000 x < 4000
Setting up the Equations for the Fraction Now, let's find the fraction. Let the fraction be b a . According to the problem, adding 2 to both the numerator and the denominator gives us 10 9 , and subtracting 3 from both the numerator and the denominator gives us 5 4 . This gives us the following system of equations: b + 2 a + 2 = 10 9 b − 3 a − 3 = 5 4
Cross-Multiplying We can cross-multiply to eliminate the fractions: 10 ( a + 2 ) = 9 ( b + 2 ) 5 ( a − 3 ) = 4 ( b − 3 )
Expanding the Equations Expanding these equations, we get: 10 a + 20 = 9 b + 18 5 a − 15 = 4 b − 12
Rearranging Terms Rearranging the terms, we have: 10 a − 9 b = − 2 5 a − 4 b = 3
Multiplying the Second Equation To solve this system of equations, we can multiply the second equation by 2: 2 ( 5 a − 4 b ) = 2 ( 3 ) 10 a − 8 b = 6
Subtracting the Equations Now we can subtract the first equation from the modified second equation: ( 10 a − 8 b ) − ( 10 a − 9 b ) = 6 − ( − 2 ) b = 8
Solving for a Substitute b = 8 into one of the equations, for example, 5 a − 4 b = 3 :
5 a − 4 ( 8 ) = 3 5 a − 32 = 3 5 a = 35 a = 7
Finding the Fraction Therefore, the fraction is b a = 8 7 .
Final Answer In summary, the range of values for x is x < 4000 , and the fraction is 8 7 .
Examples
Understanding inequalities helps in managing budgets. For instance, if you want to spend less than $20 , 000 on a car and you know you'll be paying $5 per mile in gas, you can use the inequality 5 x < 20000 to determine the maximum number of miles you can drive to stay within your budget. Solving systems of equations is useful in chemistry when balancing chemical equations or in finance when determining the optimal mix of investments to maximize returns while minimizing risk. For example, if you know the total cost and the relative prices of two ingredients in a mixture, you can set up a system of equations to find the exact quantities of each ingredient.
