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Sagot :
To determine the solution to the equation [tex]\( x + 5 = -3^x + 4 \)[/tex] to the nearest fourth of a unit, let's follow a detailed, step-by-step approach using a table of values.
Step 1: Understanding the Equation
The equation we need to solve is:
[tex]\[ x + 5 = -3^x + 4 \][/tex]
Step 2: Simplify and Rearrange
We can rearrange the equation to have all terms on one side:
[tex]\[ x + 5 - 4 = -3^x \][/tex]
[tex]\[ x + 1 = -3^x \][/tex]
Step 3: Create a Table of Values
We'll evaluate both sides of the equation for various values of [tex]\( x \)[/tex] close to the provided options.
Let's evaluate for [tex]\( x \)[/tex] values around the given options: -2.25, -1.25, 1.25, and 3.75.
| [tex]\( x \)[/tex] | [tex]\( x + 1 \)[/tex] | [tex]\( -3^x \)[/tex] | Difference ([tex]\( x + 1 + 3^x \)[/tex]) |
|-----------|-------------|------------|-----------------------------|
| -2.25 | -2.25 + 1 = -1.25 | [tex]\( -3^{-2.25} \approx -0.057 \)[/tex] | [tex]\(-1.25 + 0.057 = -1.193 \)[/tex] |
| -1.25 | -1.25 + 1 = -0.25 | [tex]\( -3^{-1.25} \approx -0.18 \)[/tex] | [tex]\(-0.25 + 0.18 = -0.07 \)[/tex] |
| 1.25 | 1.25 + 1 = 2.25 | [tex]\( -3^{1.25} \approx -4.76 \)[/tex] | [tex]\( 2.25 + 4.76 = -2.51 \)[/tex] |
| 3.75 | 3.75 + 1 = 4.75 | [tex]\( -3^{3.75} \approx -112.49 \)[/tex] | [tex]\( 4.75 + 112.49 = -107.74 \)[/tex]|
Step 4: Analyze the Results
From the table, we observe the following:
- When [tex]\( x = -2.25 \)[/tex], the difference is [tex]\( -1.193 \)[/tex] (not close to zero).
- When [tex]\( x = -1.25 \)[/tex], the difference is [tex]\( -0.07 \)[/tex] (very close to zero).
- When [tex]\( x = 1.25 \)[/tex], the difference is [tex]\( -2.51 \)[/tex] (not close to zero).
- When [tex]\( x = 3.75 \)[/tex], the difference is [tex]\( -107.74 \)[/tex] (not close to zero).
The value that brings the right-hand side and the left-hand side of the equation closest to equality is [tex]\( x \approx -1.25 \)[/tex].
Step 5: Conclusion
Based on the table of values and our analysis, the correct answer is:
[tex]\[ \boxed{A. \, x \approx -1.25} \][/tex]
Step 1: Understanding the Equation
The equation we need to solve is:
[tex]\[ x + 5 = -3^x + 4 \][/tex]
Step 2: Simplify and Rearrange
We can rearrange the equation to have all terms on one side:
[tex]\[ x + 5 - 4 = -3^x \][/tex]
[tex]\[ x + 1 = -3^x \][/tex]
Step 3: Create a Table of Values
We'll evaluate both sides of the equation for various values of [tex]\( x \)[/tex] close to the provided options.
Let's evaluate for [tex]\( x \)[/tex] values around the given options: -2.25, -1.25, 1.25, and 3.75.
| [tex]\( x \)[/tex] | [tex]\( x + 1 \)[/tex] | [tex]\( -3^x \)[/tex] | Difference ([tex]\( x + 1 + 3^x \)[/tex]) |
|-----------|-------------|------------|-----------------------------|
| -2.25 | -2.25 + 1 = -1.25 | [tex]\( -3^{-2.25} \approx -0.057 \)[/tex] | [tex]\(-1.25 + 0.057 = -1.193 \)[/tex] |
| -1.25 | -1.25 + 1 = -0.25 | [tex]\( -3^{-1.25} \approx -0.18 \)[/tex] | [tex]\(-0.25 + 0.18 = -0.07 \)[/tex] |
| 1.25 | 1.25 + 1 = 2.25 | [tex]\( -3^{1.25} \approx -4.76 \)[/tex] | [tex]\( 2.25 + 4.76 = -2.51 \)[/tex] |
| 3.75 | 3.75 + 1 = 4.75 | [tex]\( -3^{3.75} \approx -112.49 \)[/tex] | [tex]\( 4.75 + 112.49 = -107.74 \)[/tex]|
Step 4: Analyze the Results
From the table, we observe the following:
- When [tex]\( x = -2.25 \)[/tex], the difference is [tex]\( -1.193 \)[/tex] (not close to zero).
- When [tex]\( x = -1.25 \)[/tex], the difference is [tex]\( -0.07 \)[/tex] (very close to zero).
- When [tex]\( x = 1.25 \)[/tex], the difference is [tex]\( -2.51 \)[/tex] (not close to zero).
- When [tex]\( x = 3.75 \)[/tex], the difference is [tex]\( -107.74 \)[/tex] (not close to zero).
The value that brings the right-hand side and the left-hand side of the equation closest to equality is [tex]\( x \approx -1.25 \)[/tex].
Step 5: Conclusion
Based on the table of values and our analysis, the correct answer is:
[tex]\[ \boxed{A. \, x \approx -1.25} \][/tex]
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