Consider the equation and the graph.
[tex]\frac{3}{x+4}=5+1[/tex]
-37/16
-30/16
[tex]$-33 / 16$[/tex]
-35/16
The approximate solution to the given equation after three iterations of successive approximations is when [tex]x[/tex] is about
Answer (1)
Simplify the equation: x + 4 3 = 6 .
Isolate x : x = 2 1 − 4 = 2 − 7 = 16 − 56 .
Compare the calculated value with the given options.
The closest option to the actual solution x = − 56/16 is − 16 35 .
Explanation
Problem Analysis We are given the equation x + 4 3 = 5 + 1 and a set of possible solutions for x : -37/16, -30/16, -33/16, and -35/16. Our goal is to find which of these values is the approximate solution to the equation.
Simplifying the Equation First, simplify the right-hand side of the equation: 5 + 1 = 6 . So the equation becomes x + 4 3 = 6 .
Isolating x Next, we want to isolate x . Multiply both sides of the equation by ( x + 4 ) to get 3 = 6 ( x + 4 ) .
Further Isolation Divide both sides by 6 to get 6 3 = x + 4 , which simplifies to 2 1 = x + 4 .
Isolating x Subtract 4 from both sides to isolate x : x = 2 1 − 4 .
Fraction Conversion Convert 4 to a fraction with a denominator of 2: 4 = 2 8 .
Calculating x Calculate x = 2 1 − 2 8 = 2 1 − 8 = 2 − 7 .
Converting to Common Denominator Convert the result to a fraction with a denominator of 16: x = 2 − 7 = 2 × 8 − 7 × 8 = 16 − 56 .
Finding the Closest Option Now, let's compare our calculated value x = 16 − 56 with the given options: -37/16, -30/16, -33/16, -35/16. We are looking for the closest value. However, our calculated value is not in the options. It seems there was a misunderstanding and the question is asking which of the given options is closest to the solution of the equation x + 4 3 = 6 . The correct solution is x = 16 − 56 . We can check which of the options is closest to 6 when plugged into the original equation.
Let's test each option:
For x = − 37/16 : ( − 37/16 ) + 4 3 = ( − 37/16 ) + ( 64/16 ) 3 = ( 27/16 ) 3 = 27 3 × 16 = 9 16 ≈ 1.78 For x = − 30/16 : ( − 30/16 ) + 4 3 = ( − 30/16 ) + ( 64/16 ) 3 = ( 34/16 ) 3 = 34 3 × 16 = 34 48 = 17 24 ≈ 1.41 For x = − 33/16 : ( − 33/16 ) + 4 3 = ( − 33/16 ) + ( 64/16 ) 3 = ( 31/16 ) 3 = 31 3 × 16 = 31 48 ≈ 1.55 For x = − 35/16 : ( − 35/16 ) + 4 3 = ( − 35/16 ) + ( 64/16 ) 3 = ( 29/16 ) 3 = 29 3 × 16 = 29 48 ≈ 1.66
Since we want the expression to equal 6, we need to find which x makes x + 4 3 closest to 6. The correct solution to the equation is x = − 7/2 = − 56/16 . However, we need to choose from the given options. The closest value to -56/16 is -35/16. Let's check the value of the expression for x = − 35/16 . x + 4 3 = ( − 35/16 ) + 4 3 = ( − 35/16 ) + ( 64/16 ) 3 = ( 29/16 ) 3 = 29 48 ≈ 1.66 . Since the equation should equal 6, we are looking for the value of x that makes the expression equal to 6. The closest value to the actual solution is -35/16.
Final Answer The closest option to the actual solution x = − 56/16 is − 35/16 .
Examples
Understanding how to solve equations like this is crucial in many fields, such as physics and engineering. For example, if you're designing a circuit and need to determine the resistance needed to achieve a specific current, you might end up with an equation similar to the one we solved. By manipulating the equation and isolating the variable, you can find the exact resistance required. This skill is also valuable in economics, where you might need to determine the price point at which supply equals demand.
