To solve the equation [tex]\(2|x-2|-5=\sqrt{x+3}-1\)[/tex] graphically, we can break it down into two parts and plot each part on a graph:
1. Left Side of the Equation: [tex]\(2|x-2|-5\)[/tex]
- The term [tex]\( |x-2| \)[/tex] is the absolute value function shifted right by 2 units.
- When multiplied by 2, it stretches vertically by a factor of 2.
- Subtracting 5 shifts the entire function downward by 5 units.
2. Right Side of the Equation: [tex]\(\sqrt{x+3}-1\)[/tex]
- The term [tex]\(\sqrt{x+3}\)[/tex] is the square root function shifted left by 3 units.
- Subtracting 1 shifts the entire function downward by 1 unit.
Next, let's see what their intersection points might be. These intersections represent solutions of the equation [tex]\(2|x-2|-5=\sqrt{x+3}-1\)[/tex]. Using graphical methods, we determine where the two curves intersect graphically.
### Step-by-Step Solution:
#### Step 1: Analyze and Sketch the Graphs
1. Plot [tex]\(2|x-2|-5\)[/tex]:
- Start with [tex]\(y = |x-2|\)[/tex], a V-shaped graph with a vertex at [tex]\((2,0)\)[/tex].
- Stretch this by a factor of 2 to get [tex]\(y = 2|x-2|\)[/tex].
- Shift the graph downward by 5 units to get [tex]\(y = 2|x-2| - 5\)[/tex]. This will have a vertex at [tex]\((2,-5)\)[/tex].
2. Plot [tex]\(\sqrt{x+3} - 1\)[/tex]:
- Start with [tex]\(y=\sqrt{x}\)[/tex], a curve starting from the origin [tex]\((0,0)\)[/tex].
- Shift this left by 3 units to get [tex]\(y=\sqrt{x+3}\)[/tex].
- Shift this downward by 1 unit to get [tex]\(y=\sqrt{x+3} - 1\)[/tex]. The new starting point is [tex]\((-3,-1)\)[/tex].
#### Step 2: Find Intersection Points Graphically
Graphing these two functions on the same coordinate plane can help visually identify the points of intersection. These points are the approximate solutions to the equation.
From the graphs:
1. The functions intersect roughly at [tex]\(x \approx -0.50\)[/tex].
2. The functions intersect again roughly at [tex]\(x \approx 2\)[/tex].
### Final Solution:
Thus, the approximate solutions to the equation are:
[tex]\[ x \approx -0.50 \quad \text{and} \quad x \approx 2 \][/tex]
### Conclusion:
The correct option is:
C. [tex]\(x \approx -0.50\)[/tex] and [tex]\(x \approx 2\)[/tex].
