Question

graph the compound inequality on the number line. ( x > -4 ) and ( x leq 1 ) number line with -10 to 10 and graphing tools

Explanation:

Step1: Analyze \( x > -4 \)

For \( x > -4 \), we use an open circle at \( -4 \) (since \( x \) is not equal to \( -4 \)) and draw an arrow to the right from \( -4 \) to represent all numbers greater than \( -4 \).

Step2: Analyze \( x \leq 1 \)

For \( x \leq 1 \), we use a closed circle at \( 1 \) (since \( x \) can be equal to \( 1 \)) and draw an arrow to the left from \( 1 \) to represent all numbers less than or equal to \( 1 \).

Step3: Find the intersection

The compound inequality \( x > -4 \) and \( x \leq 1 \) represents the intersection of the two sets. So we combine the two graphs: an open circle at \( -4 \), a closed circle at \( 1 \), and a line segment connecting them (since the solution is all numbers greater than \( -4 \) and less than or equal to \( 1 \)).

To graph this:

1. On the number line, place an open circle at \( -4 \) (because \( x > -4 \), not \( x \geq -4 \)).
2. Place a closed circle at \( 1 \) (because \( x \leq 1 \), so \( 1 \) is included).
3. Draw a line segment connecting the open circle at \( -4 \) and the closed circle at \( 1 \), with arrows on the segment (or just the segment between them) to show the solution set \( -4 < x \leq 1 \).

Answer:

To graph \( x > -4 \) and \( x \leq 1 \):

• At \( x = -4 \), draw an open circle (since \( x > -4 \), \( -4 \) is not included) and draw a line to the right from \( -4 \) (but limited by the other inequality).
• At \( x = 1 \), draw a closed circle (since \( x \leq 1 \), \( 1 \) is included) and draw a line to the left from \( 1 \).
• The solution is the segment between \( -4 \) (open circle) and \( 1 \) (closed circle), representing \( -4 < x \leq 1 \).

(Visually, on the given number line: mark an open circle at \( -4 \), a closed circle at \( 1 \), and shade the region between them.)