Question

graphing inequalities
graph the inequality:

1. ( x geq -2 )
2. ( x < 5 )
3. ( x leq 6 )
4. ( x > 1 )
5. ( x < -14 )
6. ( x leq ) (partially visible)
7. ( x > -2 )
8. (partially visible)
9. ( x leq -10 )
10. (partially visible)

Explanation:

Let's take problem 1: \( x \geq -2 \) as an example to show the graphing steps.

Step1: Identify the critical point

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

Step2: Determine the type of dot

Since the inequality is "greater than or equal to", we use a closed (filled) dot at \( x = -2 \) to indicate that -2 is included in the solution set.

Step3: Determine the direction of the line

For \( x \geq -2 \), the solutions are all numbers greater than or equal to -2, so we draw a line to the right of the closed dot at \( x = -2 \) on the number line.

For problem 2: \( x < 5 \)

Step1: Identify the critical point

The critical point is \( x = 5 \).

Step2: Determine the type of dot

Since the inequality is "less than" (not "less than or equal to"), we use an open (unfilled) dot at \( x = 5 \) to indicate that 5 is not included in the solution set.

Step3: Determine the direction of the line

For \( x < 5 \), the solutions are all numbers less than 5, so we draw a line to the left of the open dot at \( x = 5 \) on the number line.

For problem 3: \( x \leq 6 \)

Step1: Identify the critical point

The critical point is \( x = 6 \).

Step2: Determine the type of dot

Since the inequality is "less than or equal to", we use a closed dot at \( x = 6 \).

Step3: Determine the direction of the line

For \( x \leq 6 \), the solutions are all numbers less than or equal to 6, so we draw a line to the left of the closed dot at \( x = 6 \) on the number line (the given number line has marks from -12 to 12, so we shade left from 6).

For problem 4: \( x > 1 \) (assuming the number line has appropriate marks)

Step1: Identify the critical point

The critical point is \( x = 1 \).

Step2: Determine the type of dot

Since the inequality is "greater than", we use an open dot at \( x = 1 \).

Step3: Determine the direction of the line

For \( x > 1 \), the solutions are all numbers greater than 1, so we draw a line to the right of the open dot at \( x = 1 \).

For problem 5: \( x < -14 \)

Step1: Identify the critical point

The critical point is \( x = -14 \).

Step2: Determine the type of dot

Since the inequality is "less than", we use an open dot at \( x = -14 \).

Step3: Determine the direction of the line

For \( x < -14 \), the solutions are all numbers less than -14, so we draw a line to the left of the open dot at \( x = -14 \) on the number line (marks from -42 to 42, so shade left from -14).

For problem 6: (assuming it's \( x \leq \) some number, say if it's \( x \leq -15 \) with number line from -18 etc.)

Step1: Identify the critical point

Suppose the critical point is \( x = -15 \) (depending on the actual problem, but following the pattern).

Step2: Determine the type of dot

Closed dot if "less than or equal to".

Step3: Determine the direction of the line

Draw line to the left of the closed dot.

For problem 7: \( x > -2 \)

Step1: Identify the critical point

The critical point is \( x = -2 \).

Step2: Determine the type of dot

Open dot (since it's "greater than", not "greater than or equal to").

Step3: Determine the direction of the line

Draw a line to the right of the open dot at \( x = -2 \) on the number line (marks from -12 to 12, so shade right from -2).

For problem 8: (assuming it's a similar inequality, say \( x < -30 \) with number line from -30 etc.)

Step1: Identify the critical point

Critical point at \( x = -30 \).

Step2: Determine the type of dot

Open dot (if "less than").

Step3: Determine the direction of the line

Draw line to the left of the open dot.

For problem 9: \( x \leq -10 \)

Step1: Identify the critical point

The critica…

Answer:

The graphs are drawn by:

1. For \( x \geq -2 \): Closed dot at -2, line to the right.
2. For \( x < 5 \): Open dot at 5, line to the left.
3. For \( x \leq 6 \): Closed dot at 6, line to the left.
4. For \( x > 1 \): Open dot at 1, line to the right.
5. For \( x < -14 \): Open dot at -14, line to the left.
6. (Depends on the full inequality) Follow critical point, dot type, direction.
7. For \( x > -2 \): Open dot at -2, line to the right.
8. (Depends on the full inequality) Follow critical point, dot type, direction.
9. For \( x \leq -10 \): Closed dot at -10, line to the left.
10. (Depends on the full inequality) Follow critical point, dot type, direction.