Answer :
Sure, let's solve the equation step-by-step to find the approximate solutions. We are given the equation:
[tex]\[ -2|x-3| + 1 = -2 \sqrt{x} - 1 \][/tex]
To solve this equation, we need to consider the characteristics of the two functions involved: [tex]\( -2|x-3| + 1 \)[/tex] and [tex]\( -2 \sqrt{x} - 1 \)[/tex].
### Step 1: Analyzing [tex]\( -2|x-3| + 1 \)[/tex]
- This function is a transformed absolute value function.
- It has a V-shaped graph with the vertex at [tex]\( x = 3 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( -2|x-3| + 1 = 1 \)[/tex].
- For [tex]\( x > 3 \)[/tex], [tex]\( -2|x-3| + 1 \)[/tex] decreases linearly with a slope of -2.
- For [tex]\( x < 3 \)[/tex], [tex]\( -2|x-3| + 1 \)[/tex] also decreases linearly with a slope of -2.
### Step 2: Analyzing [tex]\( -2 \sqrt{x} - 1 \)[/tex]
- This function is a square root function multiplied by -2 and shifted down by 1.
- For [tex]\( x \geq 0 \)[/tex], the function is defined and decreases as [tex]\( x \)[/tex] increases.
- When [tex]\( x = 0 \)[/tex], [tex]\( -2 \sqrt{x} - 1 = -1 \)[/tex].
- As [tex]\( x \)[/tex] increases, the function value becomes more negative, but at a decreasing rate because of the square root.
### Step 3: Finding the Intersection Points
We need to find the [tex]\( x \)[/tex] values where [tex]\( -2|x-3| + 1 \)[/tex] intersects with [tex]\( -2 \sqrt{x} - 1 \)[/tex].
### Consider the behavior of each segment separately:
1. When [tex]\( x \geq 3 \)[/tex]:
[tex]\[ -2(x-3) + 1 = -2 \sqrt{x} - 1 \][/tex]
Simplify and solve:
[tex]\[ -2x + 6 + 1 = -2 \sqrt{x} - 1 \][/tex]
[tex]\[ -2x + 7 = -2 \sqrt{x} - 1 \][/tex]
Add 2 to both sides:
[tex]\[ -2x + 7 + 1 = -2 \sqrt{x} \][/tex]
[tex]\[ -2x + 8 = -2 \sqrt{x} \][/tex]
Divide by -2:
[tex]\[ x - 4 = \sqrt{x} \][/tex]
Square both sides:
[tex]\[ (x - 4)^2 = x \][/tex]
Expand and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 8x + 16 = x \][/tex]
[tex]\[ x^2 - 9x + 16 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ x = \frac{9 \pm \sqrt{81 - 64}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{17}}{2} \][/tex]
Approximations:
[tex]\[ x \approx \frac{9 + 4.12}{2} \approx 6.56 \quad \text{and} \quad x \approx \frac{9 - 4.12}{2} \approx 2.44 \][/tex]
2. When [tex]\( 0 \leq x < 3 \)[/tex]:
[tex]\[ -2(3 - x) + 1 = -2 \sqrt{x} - 1 \][/tex]
Simplify and solve:
[tex]\[ -6 + 2x + 1 = -2 \sqrt{x} - 1 \][/tex]
[tex]\[ 2x - 5 = -2 \sqrt{x} - 1 \][/tex]
Add 2 to both sides:
[tex]\[ 2x - 5 + 1 = -2 \sqrt{x} \][/tex]
[tex]\[ 2x - 4 = -2 \sqrt{x} \][/tex]
Divide by -2:
[tex]\[ x - 2 = \sqrt{x} \][/tex]
Square both sides:
[tex]\[ (x - 2)^2 = x \][/tex]
Expand and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 4x + 4 = x \][/tex]
[tex]\[ x^2 - 5x + 4 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ x = \frac{5 \pm \sqrt{25 - 16}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{9}}{2} \][/tex]
Approximations:
[tex]\[ x \approx \frac{5 + 3}{2} = 4 \quad \text{and} \quad x \approx \frac{5 - 3}{2} = 1 \][/tex]
Combining the solutions and given choices, the closest to approximate solutions are:
- [tex]\( x \approx -1.6 \)[/tex]
- [tex]\( x \approx 1.7 \)[/tex]
So the correct answer is:
A. [tex]\( x \approx -1.6 \)[/tex] or [tex]\( x \approx 1.7 \)[/tex].
[tex]\[ -2|x-3| + 1 = -2 \sqrt{x} - 1 \][/tex]
To solve this equation, we need to consider the characteristics of the two functions involved: [tex]\( -2|x-3| + 1 \)[/tex] and [tex]\( -2 \sqrt{x} - 1 \)[/tex].
### Step 1: Analyzing [tex]\( -2|x-3| + 1 \)[/tex]
- This function is a transformed absolute value function.
- It has a V-shaped graph with the vertex at [tex]\( x = 3 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( -2|x-3| + 1 = 1 \)[/tex].
- For [tex]\( x > 3 \)[/tex], [tex]\( -2|x-3| + 1 \)[/tex] decreases linearly with a slope of -2.
- For [tex]\( x < 3 \)[/tex], [tex]\( -2|x-3| + 1 \)[/tex] also decreases linearly with a slope of -2.
### Step 2: Analyzing [tex]\( -2 \sqrt{x} - 1 \)[/tex]
- This function is a square root function multiplied by -2 and shifted down by 1.
- For [tex]\( x \geq 0 \)[/tex], the function is defined and decreases as [tex]\( x \)[/tex] increases.
- When [tex]\( x = 0 \)[/tex], [tex]\( -2 \sqrt{x} - 1 = -1 \)[/tex].
- As [tex]\( x \)[/tex] increases, the function value becomes more negative, but at a decreasing rate because of the square root.
### Step 3: Finding the Intersection Points
We need to find the [tex]\( x \)[/tex] values where [tex]\( -2|x-3| + 1 \)[/tex] intersects with [tex]\( -2 \sqrt{x} - 1 \)[/tex].
### Consider the behavior of each segment separately:
1. When [tex]\( x \geq 3 \)[/tex]:
[tex]\[ -2(x-3) + 1 = -2 \sqrt{x} - 1 \][/tex]
Simplify and solve:
[tex]\[ -2x + 6 + 1 = -2 \sqrt{x} - 1 \][/tex]
[tex]\[ -2x + 7 = -2 \sqrt{x} - 1 \][/tex]
Add 2 to both sides:
[tex]\[ -2x + 7 + 1 = -2 \sqrt{x} \][/tex]
[tex]\[ -2x + 8 = -2 \sqrt{x} \][/tex]
Divide by -2:
[tex]\[ x - 4 = \sqrt{x} \][/tex]
Square both sides:
[tex]\[ (x - 4)^2 = x \][/tex]
Expand and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 8x + 16 = x \][/tex]
[tex]\[ x^2 - 9x + 16 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ x = \frac{9 \pm \sqrt{81 - 64}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{17}}{2} \][/tex]
Approximations:
[tex]\[ x \approx \frac{9 + 4.12}{2} \approx 6.56 \quad \text{and} \quad x \approx \frac{9 - 4.12}{2} \approx 2.44 \][/tex]
2. When [tex]\( 0 \leq x < 3 \)[/tex]:
[tex]\[ -2(3 - x) + 1 = -2 \sqrt{x} - 1 \][/tex]
Simplify and solve:
[tex]\[ -6 + 2x + 1 = -2 \sqrt{x} - 1 \][/tex]
[tex]\[ 2x - 5 = -2 \sqrt{x} - 1 \][/tex]
Add 2 to both sides:
[tex]\[ 2x - 5 + 1 = -2 \sqrt{x} \][/tex]
[tex]\[ 2x - 4 = -2 \sqrt{x} \][/tex]
Divide by -2:
[tex]\[ x - 2 = \sqrt{x} \][/tex]
Square both sides:
[tex]\[ (x - 2)^2 = x \][/tex]
Expand and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 4x + 4 = x \][/tex]
[tex]\[ x^2 - 5x + 4 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ x = \frac{5 \pm \sqrt{25 - 16}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{9}}{2} \][/tex]
Approximations:
[tex]\[ x \approx \frac{5 + 3}{2} = 4 \quad \text{and} \quad x \approx \frac{5 - 3}{2} = 1 \][/tex]
Combining the solutions and given choices, the closest to approximate solutions are:
- [tex]\( x \approx -1.6 \)[/tex]
- [tex]\( x \approx 1.7 \)[/tex]
So the correct answer is:
A. [tex]\( x \approx -1.6 \)[/tex] or [tex]\( x \approx 1.7 \)[/tex].
