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Sagot :
Let's break down the given inequality [tex]\( y - 5 > 2x - 10 \)[/tex] and analyze it step-by-step to graph it correctly and then identify the matching graph among the options provided.
### Step-by-Step Solution
1. Isolate [tex]\( y \)[/tex] in the inequality:
[tex]\[ y - 5 > 2x - 10 \][/tex]
Add 5 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y > 2x - 10 + 5 \][/tex]
Simplify the right-hand side:
[tex]\[ y > 2x - 5 \][/tex]
2. Rewrite the inequality in slope-intercept form (where [tex]\( y > mx + b \)[/tex]):
[tex]\[ y > 2x - 5 \][/tex]
This tells us that the boundary line of the inequality is [tex]\( y = 2x - 5 \)[/tex], and the inequality indicates that we need the region above this line (since [tex]\( y \)[/tex] is greater than [tex]\( 2x - 5 \)[/tex]).
3. Graph the boundary line [tex]\( y = 2x - 5 \)[/tex]:
- The line [tex]\( y = 2x - 5 \)[/tex] has a slope of 2 and a y-intercept of -5.
- To graph this, start at the y-intercept (0, -5). From this point, use the slope to determine another point on the line. Since the slope is 2, go up 2 units and 1 unit to the right to find another point, for example, (1, -3).
Note that the boundary line itself [tex]\( y = 2x - 5 \)[/tex] will be represented by a dashed line because the inequality is strict (greater than but not equal to).
4. Shade the appropriate region:
- Since the inequality is [tex]\( y > 2x - 5 \)[/tex], we shade the region above the line.
### Summary of the Graph:
- Dashed line representing [tex]\( y = 2x - 5 \)[/tex].
- Shaded region above the dashed line.
### Matching Graph:
- Now, you need to look at the provided graph options (Graph A, Graph B, Graph C, and Graph D) and identify which one matches the description of having a dashed line for [tex]\( y = 2x - 5 \)[/tex] with the region above this line shaded.
As this is a text-based solution and I don't have the actual graphs to look at, make sure to check the key points:
- Dashed line through y-intercept (-5) and with slope 2.
- Shaded region should be above the dashed line.
Identify the graph that matches these criteria. The correct choice among the options (B, C, or D) will depend on visual confirmation using the steps outlined above.
### Step-by-Step Solution
1. Isolate [tex]\( y \)[/tex] in the inequality:
[tex]\[ y - 5 > 2x - 10 \][/tex]
Add 5 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y > 2x - 10 + 5 \][/tex]
Simplify the right-hand side:
[tex]\[ y > 2x - 5 \][/tex]
2. Rewrite the inequality in slope-intercept form (where [tex]\( y > mx + b \)[/tex]):
[tex]\[ y > 2x - 5 \][/tex]
This tells us that the boundary line of the inequality is [tex]\( y = 2x - 5 \)[/tex], and the inequality indicates that we need the region above this line (since [tex]\( y \)[/tex] is greater than [tex]\( 2x - 5 \)[/tex]).
3. Graph the boundary line [tex]\( y = 2x - 5 \)[/tex]:
- The line [tex]\( y = 2x - 5 \)[/tex] has a slope of 2 and a y-intercept of -5.
- To graph this, start at the y-intercept (0, -5). From this point, use the slope to determine another point on the line. Since the slope is 2, go up 2 units and 1 unit to the right to find another point, for example, (1, -3).
Note that the boundary line itself [tex]\( y = 2x - 5 \)[/tex] will be represented by a dashed line because the inequality is strict (greater than but not equal to).
4. Shade the appropriate region:
- Since the inequality is [tex]\( y > 2x - 5 \)[/tex], we shade the region above the line.
### Summary of the Graph:
- Dashed line representing [tex]\( y = 2x - 5 \)[/tex].
- Shaded region above the dashed line.
### Matching Graph:
- Now, you need to look at the provided graph options (Graph A, Graph B, Graph C, and Graph D) and identify which one matches the description of having a dashed line for [tex]\( y = 2x - 5 \)[/tex] with the region above this line shaded.
As this is a text-based solution and I don't have the actual graphs to look at, make sure to check the key points:
- Dashed line through y-intercept (-5) and with slope 2.
- Shaded region should be above the dashed line.
Identify the graph that matches these criteria. The correct choice among the options (B, C, or D) will depend on visual confirmation using the steps outlined above.
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