Question

graph the follow statements. use a highlighter in shading the solution set.

1. $x \leq -6$
2. $x \geq 3$
3. $x > -2$
4. $x < 1$
5. $x \geq 5$
6. $x < -4$
7. $x > 24$
8. $x \geq -150$
9. $x < 39$
10. $x \leq 9$

Answer:

1) Graph of \( x \leq -6 \)

Step 1: Identify the critical point

The inequality is \( x \leq -6 \), so the critical point is \( -6 \).

Step 2: Determine the type of circle

Since the inequality is "less than or equal to", we use a closed (filled) circle at \( -6 \).

Step 3: Determine the direction of shading

We shade to the left of \( -6 \) (towards more negative numbers) because \( x \) is less than or equal to \( -6 \). Looking at the number line provided (with labels -3, -4, -5, -6, -7, -8, -9, -10, -11; note: the numbering seems reversed, but we follow the inequality), we shade from \( -6 \) (closed circle) to the left (towards -7, -8, etc.).

2) Graph of \( x \geq 3 \)

Step 1: Identify the critical point

The inequality is \( x \geq 3 \), so the critical point is \( 3 \).

Step 2: Determine the type of circle

Since the inequality is "greater than or equal to", we use a closed (filled) circle at \( 3 \).

Step 3: Determine the direction of shading

We shade to the right of \( 3 \) (towards larger numbers) because \( x \) is greater than or equal to \( 3 \). On the number line (labels -3, -2, -1, 0, 1, 2, 3, 4, 5), we shade from \( 3 \) (closed circle) to the right (towards 4, 5, etc.).

3) Graph of \( x > -2 \)

Step 1: Identify the critical point

The inequality is \( x > -2 \), so the critical point is \( -2 \).

Step 2: Determine the type of circle

Since the inequality is "greater than" (not equal to), we use an open (unfilled) circle at \( -2 \).

Step 3: Determine the direction of shading

We shade to the right of \( -2 \) (towards larger numbers) because \( x \) is greater than \( -2 \). On the number line (labels -3, -2, -1, 0, 1, 2, 3, 4, 5), we shade from \( -2 \) (open circle) to the right (towards -1, 0, 1, etc.).

4) Graph of \( x < 1 \)

Step 1: Identify the critical point

The inequality is \( x < 1 \), so the critical point is \( 1 \).

Step 2: Determine the type of circle

Since the inequality is "less than" (not equal to), we use an open (unfilled) circle at \( 1 \).

Step 3: Determine the direction of shading

We shade to the left of \( 1 \) (towards smaller numbers) because \( x \) is less than \( 1 \). On the number line (labels -3, -2, -1, 0, 1, 2, 3, 4, 5), we shade from \( 1 \) (open circle) to the left (towards 0, -1, -2, etc.).

5) Graph of \( x \geq 5 \)

Step 1: Identify the critical point

The inequality is \( x \geq 5 \), so the critical point is \( 5 \).

Step 2: Determine the type of circle

Since the inequality is "greater than or equal to", we use a closed (filled) circle at \( 5 \).

Step 3: Determine the direction of shading

We shade to the right of \( 5 \) (towards larger numbers) because \( x \) is greater than or equal to \( 5 \). On the number line (labels 0, 1, 2, 3, 4, 5, 6, 7, 8), we shade from \( 5 \) (closed circle) to the right (towards 6, 7, etc.).

6) Graph of \( x < -4 \)

Step 1: Identify the critical point

The inequality is \( x < -4 \), so the critical point is \( -4 \).

Step 2: Determine the type of circle

Since the inequality is "less than" (not equal to), we use an open (unfilled) circle at \( -4 \).

Step 3: Determine the direction of shading

We shade to the left of \( -4 \) (towards more negative numbers) because \( x \) is less than \( -4 \). On the number line (labels -7, -6, -5, -4, -3, -2, -1, 0, 1), we shade from \( -4 \) (open circle) to the left (towards -5, -6, -7, etc.).

7) Graph of \( x > 24 \)

Step 1: Identify the critical point

The inequality is \( x > 24 \), so the critical point is \( 24 \).

Step 2: Determine the type of circle

Since the inequality is "greater than" (not equal to), we use an open (unfilled) circle at \( 24 \).

Step 3: Determine the direction of shading

We shade to the right of \( 24 \) (towards larger numbers) because \( x \) is greater than \( 24 \). On the number line (labels 19, 20, 21, 22, 23, 24, 25, 26, 27), we shade from \( 24 \) (open circle) to the right (towards 25, 26, etc.).

8) Graph of \( x \geq -150 \)

Step 1: Identify the critical point

The inequality is \( x \geq -150 \), so the critical point is \( -150 \).

Step 2: Determine the type of circle

Since the inequality is "greater than or equal to", we use a closed (filled) circle at \( -150 \).

Step 3: Determine the direction of shading

We shade to the right of \( -150 \) (towards larger numbers) because \( x \) is greater than or equal to \( -150 \). On the number line (labels -155, -154, -153, -152, -151, -150, -149, -148, -147), we shade from \( -150 \) (closed circle) to the right (towards -149, -148, etc.).

9) Graph of \( x < 39 \)

Step 1: Identify the critical point

The inequality is \( x < 39 \), so the critical point is \( 39 \).

Step 2: Determine the type of circle

Since the inequality is "less than" (not equal to), we use an open (unfilled) circle at \( 39 \).

Step 3: Determine the direction of shading

We shade to the left of \( 39 \) (towards smaller numbers) because \( x \) is less than \( 39 \). On the number line (labels 31, 32, 33, 34, 35, 36, 37, 38, 39), we shade from \( 39 \) (open circle) to the left (towards 38, 37, etc.).

10) Graph of \( x \leq 9 \)

Step 1: Identify the critical point

The inequality is \( x \leq 9 \), so the critical point is \( 9 \).

Step 2: Determine the type of circle

Since the inequality is "less than or equal to", we use a closed (filled) circle at \( 9 \).

Step 3: Determine the direction of shading

We shade to the left of \( 9 \) (towards smaller numbers) because \( x \) is less than or equal to \( 9 \). On the number line (labels 5, 6, 7, 8, 9, 10, 11, 12, 13), we shade from \( 9 \) (closed circle) to the left (towards 8, 7, etc.).

(Note: For actual graphing, use a highlighter to shade the regions as described. The key is to identify the critical point, the type of circle (open or closed), and the direction of shading based on the inequality symbol.)