Question

graph given solution set below
$x > -1$ and $x \leq -5$

write compound inequality statement for given
number line below.

Explanation:

Step1: Analyze the first compound inequality \(x > - 1\) and \(x\leq - 5\)

For \(x > - 1\), we draw an open circle at \(-1\) and shade to the right. For \(x\leq - 5\), we draw a closed circle at \(-5\) and shade to the left. Since there is no overlap between \(x > - 1\) (values to the right of \(-1\)) and \(x\leq - 5\) (values to the left of or equal to \(-5\)), the solution set is empty.

Step2: Analyze the number - line for writing the compound inequality

For the number - line:

• The left - hand part: We have a shaded region to the left (towards \(-\infty\)) with no specific upper - bound (except at the right - hand end of the shaded part).
• The right - hand part: There are two closed circles. Let's assume the left - hand closed circle is at \(x = 3\) (assuming a unit length of 1 between the marks) and the right - hand closed circle is at \(x = 7\). The compound inequality for a number - line with two non - overlapping parts (a left - hand ray and a right - hand segment) is of the form \(x\leq a\) or \(x\geq b\).

Answer:

1. The solution set of \(x > - 1\) and \(x\leq - 5\) is \(\varnothing\) (empty set).
2. Assuming the left - hand closed circle is at \(x = 3\) and the right - hand closed circle is at \(x = 7\), the compound inequality is \(x\leq3\) or \(x\geq7\)