Question

choose test values and indicate whether the inequality is true or false for the test values from smallest to largest.

test value:
inequality: select an answer select an answer select an answer

draw the solution above.
write the solution as a compound inequality

Explanation:

⚡ Using what you learned: Solving Compound Inequalities

Step 1: Identify the intervals and choose test values

The boundary points on the number line are \( -1 \) and \( 7 \). These points divide the number line into three distinct intervals:

1. Left interval: \( (-\infty, -1) \)
2. Middle interval: \( [-1, 7] \)
3. Right interval: \( (7, \infty) \)

We need to choose one test value (an integer or decimal) from each interval, ordered from smallest to largest:

• For the left interval \( (-\infty, -1) \), we can choose \( -2 \).
• For the middle interval \( [-1, 7] \), we can choose \( 0 \).
• For the right interval \( (7, \infty) \), we can choose \( 8 \).

Step 2: Determine the truth value of the inequality

The blue solid dots at \( -1 \) and \( 7 \) indicate that these boundary points are included in the solution set. The interval between them, \( [-1, 7] \), represents the solution region where the inequality is true.

• For the test value \( -2 \) (outside the interval): False
• For the test value \( 0 \) (inside the interval): True
• For the test value \( 8 \) (outside the interval): False

Step 3: Write the compound inequality

Since the solution lies between \( -1 \) and \( 7 \) inclusive, we write this as a compound inequality:

$$ -1 \le x \le 7 $$

Answer:

Table Values (from left to right):

• First Column:
• Test Value: -2
• Inequality: False
• Second Column:
• Test Value: 0
• Inequality: True
• Third Column:
• Test Value: 8
• Inequality: False

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Compound Inequality:

$$ -1 \le x \le 7 $$