To graph the compound inequality \( x > -6 \) and \( x \leq 7 \) on a number line, follow these steps:
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Identify the interval for \( x > -6 \):
- This means that \( x \) can take any value greater than -6. The boundary -6 is not included, so we use an open circle at -6.
- The graph will extend to the right, indicating all values greater than -6.
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Identify the interval for \( x \leq 7 \):
- This means that \( x \) can take any value less than or equal to 7. The boundary 7 is included, so we use a closed circle at 7.
- The graph will extend to the left, indicating all values less than or equal to 7.
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Combine both intervals:
- Since the inequalities must be satisfied simultaneously, we focus on the overlapping region between the two intervals, which is from just above -6 to 7.
Now, here's how you can represent this graphically on a number line:
<----(-----●==================●-----)---->
-6 7
- The open circle ( ) at -6 means that -6 is not included.
- The closed circle (●) at 7 means that 7 is included.
- The line between -6 and 7, indicated by the equals sign (=), shows all the values that are included in the solution.
Thus, the solution for the compound inequality \( x > -6 \) and \( x \leq 7 \) on the number line is represented by the interval (-6, 7].