At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To solve the equation
[tex]\[ 2|x-2| - 5 = \sqrt{x+3} - 1 \][/tex]
graphically, we'll first consider the graphs of the two sides of the equation independently:
1. [tex]\( y_1 = 2|x-2| - 5 \)[/tex]
2. [tex]\( y_2 = \sqrt{x+3} - 1 \)[/tex]
We'll look at the points of intersection of these two graphs to determine the values of [tex]\( x \)[/tex] that satisfy the equation. Let's sketch or plot these graphs:
Step 1: Graph [tex]\( y_1 = 2|x-2| - 5 \)[/tex]
- The absolute value function [tex]\( |x-2| \)[/tex] translates the graph of [tex]\( |x| \)[/tex] horizontally to the right by 2 units.
- The function [tex]\( 2|x-2| \)[/tex] then stretches this graph vertically by a factor of 2.
- Finally, subtracting 5 shifts the graph downward by 5 units.
The resulting graph forms a "V" shape, with the vertex at [tex]\( (2, -5) \)[/tex].
Step 2: Graph [tex]\( y_2 = \sqrt{x+3} - 1\)[/tex]
- The square root function [tex]\( \sqrt{x+3} \)[/tex] translates the graph of [tex]\( \sqrt{x} \)[/tex] horizontally to the left by 3 units.
- The function [tex]\( \sqrt{x+3} - 1 \)[/tex] then shifts this graph downward by 1 unit.
Step 3: Find Points of Intersection
1. From visual inspection of the graphs:
- The [tex]\( y_1 \)[/tex] function converges from two directions due to the absolute value term, creating a "V" shape with a vertex at [tex]\( (2, -5) \)[/tex].
- The [tex]\( y_2 \)[/tex] function starts at [tex]\(( -3, -1 ) \)[/tex] and increases slowly, following a square root curve shifted to the right by three units and down by one unit.
We need to find the approximate [tex]\( x \)[/tex]-values where these graphs intersect.
From the possible options, you can analyze:
- Graphically examining the [tex]\( y = 2|x-2| - 5 \)[/tex] and [tex]\( y = \sqrt{x+3} - 1 \)[/tex] curves, solutions points [tex]\( x \approx -0.50 \)[/tex] and [tex]\( x \approx 4.5 \)[/tex] commonly match the regions where these two graphs intersect, when analyzing these points in detail.
Thus, the correct answer is:
C. [tex]\( x \approx -0.50 \)[/tex] and [tex]\( x \approx 4.5 \)[/tex]
[tex]\[ 2|x-2| - 5 = \sqrt{x+3} - 1 \][/tex]
graphically, we'll first consider the graphs of the two sides of the equation independently:
1. [tex]\( y_1 = 2|x-2| - 5 \)[/tex]
2. [tex]\( y_2 = \sqrt{x+3} - 1 \)[/tex]
We'll look at the points of intersection of these two graphs to determine the values of [tex]\( x \)[/tex] that satisfy the equation. Let's sketch or plot these graphs:
Step 1: Graph [tex]\( y_1 = 2|x-2| - 5 \)[/tex]
- The absolute value function [tex]\( |x-2| \)[/tex] translates the graph of [tex]\( |x| \)[/tex] horizontally to the right by 2 units.
- The function [tex]\( 2|x-2| \)[/tex] then stretches this graph vertically by a factor of 2.
- Finally, subtracting 5 shifts the graph downward by 5 units.
The resulting graph forms a "V" shape, with the vertex at [tex]\( (2, -5) \)[/tex].
Step 2: Graph [tex]\( y_2 = \sqrt{x+3} - 1\)[/tex]
- The square root function [tex]\( \sqrt{x+3} \)[/tex] translates the graph of [tex]\( \sqrt{x} \)[/tex] horizontally to the left by 3 units.
- The function [tex]\( \sqrt{x+3} - 1 \)[/tex] then shifts this graph downward by 1 unit.
Step 3: Find Points of Intersection
1. From visual inspection of the graphs:
- The [tex]\( y_1 \)[/tex] function converges from two directions due to the absolute value term, creating a "V" shape with a vertex at [tex]\( (2, -5) \)[/tex].
- The [tex]\( y_2 \)[/tex] function starts at [tex]\(( -3, -1 ) \)[/tex] and increases slowly, following a square root curve shifted to the right by three units and down by one unit.
We need to find the approximate [tex]\( x \)[/tex]-values where these graphs intersect.
From the possible options, you can analyze:
- Graphically examining the [tex]\( y = 2|x-2| - 5 \)[/tex] and [tex]\( y = \sqrt{x+3} - 1 \)[/tex] curves, solutions points [tex]\( x \approx -0.50 \)[/tex] and [tex]\( x \approx 4.5 \)[/tex] commonly match the regions where these two graphs intersect, when analyzing these points in detail.
Thus, the correct answer is:
C. [tex]\( x \approx -0.50 \)[/tex] and [tex]\( x \approx 4.5 \)[/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.