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Sagot :
To determine which linear inequality, when combined with [tex]\( y > -x - 2 \)[/tex], creates the given solution set, let's analyze the properties of each given inequality one by one.
1. Inequality: [tex]\( y > x + 1 \)[/tex]
- This inequality represents the region above the line [tex]\( y = x + 1 \)[/tex]. We are interested in finding whether this region can overlap with the region defined by [tex]\( y > -x - 2 \)[/tex].
- By graphing these two lines:
- [tex]\( y = -x - 2 \)[/tex]: This line has a slope of -1 and a y-intercept at -2.
- [tex]\( y = x + 1 \)[/tex]: This line has a slope of 1 and a y-intercept at 1.
- The solution set for this inequality includes all points [tex]\((x, y)\)[/tex] that lie above both lines.
- [tex]\( y > -x - 2 \)[/tex] and [tex]\( y > x + 1 \)[/tex] do overlap to form a valid solution.
2. Inequality: [tex]\( y < x - 1 \)[/tex]
- This inequality represents the region below the line [tex]\( y = x - 1 \)[/tex].
- By graphing:
- [tex]\( y = x - 1 \)[/tex]: This line has a slope of 1 and a y-intercept at -1.
- The region below this line does not overlap in a significant way with the region defined by [tex]\( y > -x - 2 \)[/tex] to create the solution set. Instead, it describes a different region.
3. Inequality: [tex]\( y > x - 1 \)[/tex]
- This inequality represents the region above the line [tex]\( y = x - 1 \)[/tex].
- By graphing:
- [tex]\( y = x - 1 \)[/tex]: This line has a slope of 1 and a y-intercept at -1.
- Similar to the first case, the solution set for [tex]\( y > x - 1 \)[/tex] includes all points [tex]\((x, y)\)[/tex] above this line.
- The region above this line can definitely contribute to forming a valid overlapping solution set with [tex]\( y > -x - 2 \)[/tex].
4. Inequality: [tex]\( y < x + 1 \)[/tex]
- This inequality represents the region below the line [tex]\( y = x + 1 \)[/tex].
- By graphing:
- [tex]\( y = x + 1 \)[/tex]: This line has a slope of 1 and a y-intercept at 1.
- The region below this line does not overlap significantly with the region defined by [tex]\( y > -x - 2 \)[/tex] to contribute to the creation of the solution set.
From this analysis, we conclude that the linear inequalities [tex]\( y > x + 1 \)[/tex] and [tex]\( y > x - 1 \)[/tex] are the ones that combine with [tex]\( y > -x - 2 \)[/tex] to create the given solution set. These correspond to the first and third inequalities in the given list of options.
Therefore, the correct inequalities are:
1. [tex]\( y > x + 1 \)[/tex]
3. [tex]\( y > x - 1 \)[/tex]
In summary, the two inequalities that, when graphed with [tex]\( y > -x - 2 \)[/tex], create the given solution set are:
[tex]\[ y > x + 1 \][/tex] and [tex]\[ y > x - 1 \][/tex].
1. Inequality: [tex]\( y > x + 1 \)[/tex]
- This inequality represents the region above the line [tex]\( y = x + 1 \)[/tex]. We are interested in finding whether this region can overlap with the region defined by [tex]\( y > -x - 2 \)[/tex].
- By graphing these two lines:
- [tex]\( y = -x - 2 \)[/tex]: This line has a slope of -1 and a y-intercept at -2.
- [tex]\( y = x + 1 \)[/tex]: This line has a slope of 1 and a y-intercept at 1.
- The solution set for this inequality includes all points [tex]\((x, y)\)[/tex] that lie above both lines.
- [tex]\( y > -x - 2 \)[/tex] and [tex]\( y > x + 1 \)[/tex] do overlap to form a valid solution.
2. Inequality: [tex]\( y < x - 1 \)[/tex]
- This inequality represents the region below the line [tex]\( y = x - 1 \)[/tex].
- By graphing:
- [tex]\( y = x - 1 \)[/tex]: This line has a slope of 1 and a y-intercept at -1.
- The region below this line does not overlap in a significant way with the region defined by [tex]\( y > -x - 2 \)[/tex] to create the solution set. Instead, it describes a different region.
3. Inequality: [tex]\( y > x - 1 \)[/tex]
- This inequality represents the region above the line [tex]\( y = x - 1 \)[/tex].
- By graphing:
- [tex]\( y = x - 1 \)[/tex]: This line has a slope of 1 and a y-intercept at -1.
- Similar to the first case, the solution set for [tex]\( y > x - 1 \)[/tex] includes all points [tex]\((x, y)\)[/tex] above this line.
- The region above this line can definitely contribute to forming a valid overlapping solution set with [tex]\( y > -x - 2 \)[/tex].
4. Inequality: [tex]\( y < x + 1 \)[/tex]
- This inequality represents the region below the line [tex]\( y = x + 1 \)[/tex].
- By graphing:
- [tex]\( y = x + 1 \)[/tex]: This line has a slope of 1 and a y-intercept at 1.
- The region below this line does not overlap significantly with the region defined by [tex]\( y > -x - 2 \)[/tex] to contribute to the creation of the solution set.
From this analysis, we conclude that the linear inequalities [tex]\( y > x + 1 \)[/tex] and [tex]\( y > x - 1 \)[/tex] are the ones that combine with [tex]\( y > -x - 2 \)[/tex] to create the given solution set. These correspond to the first and third inequalities in the given list of options.
Therefore, the correct inequalities are:
1. [tex]\( y > x + 1 \)[/tex]
3. [tex]\( y > x - 1 \)[/tex]
In summary, the two inequalities that, when graphed with [tex]\( y > -x - 2 \)[/tex], create the given solution set are:
[tex]\[ y > x + 1 \][/tex] and [tex]\[ y > x - 1 \][/tex].
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