Solve the proportion.
x + 7/x + 4
=
7/x − 3
for x.
a) What is the larger answer?
b) what is the smaller answer?
To solve the given proportion `(x + 7)/(x + 4) = 7/(x - 3)` for `x`, we can start by cross-multiplying.
Step 1: Cross-multiply
`(x + 7)(x - 3) = 7(x + 4)`
Step 2: Expand and simplify
Using the distributive property:
`x^2 - 3x + 7x - 21 = 7x + 28`
Combine like terms:
`x^2 + 4x - 21 = 7x + 28`
Step 3: Simplify further
Move all the terms to one side of the equation:
`x^2 + 4x - 7x - 21 - 28 = 0`
Combine like terms:
`x^2 - 3x - 49 = 0`
Now we have a quadratic equation to solve. To find the values of `x`, we can use factoring or the quadratic formula.
Step 4: Solve the quadratic equation
Factoring `x^2 - 3x - 49 = 0`:
`(x - 7)(x + 7) = 0`
Setting each factor equal to zero:
x - 7 = 0 or x + 7 = 0
Solving for `x` in each equation:
x = 7 or x = -7
Now, let's determine which value of `x` is larger and which one is smaller.
a) To find the larger answer, we compare the values of `x`:
x = 7
b) To find the smaller answer, we compare the values of `x`:
x = -7
Therefore:
a) The larger answer is 7.
b) The smaller answer is -7.