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Which number line represents the solution set for the inequality –negative StartFraction one-half EndFraction x is greater than or equal to 4.x ≥ 4?

A number line from negative 10 to 10 in increments of 2. A point is at negative 2 and a bold line starts at negative 2 and is pointing to the left.
A number line from negative 10 to 10 in increments of 2. A point is at negative 8 and a bold line starts at negative 8 and is pointing to the left.
A number line from negative 10 to 10 in increments of 2. A point is at negative 2 and a bold line starts at negative 2 and is pointing to the right.
A number line from negative 10 to 10 in increments of 2. A point is at negative 8 and a bold line starts at negative 8 and is pointing to the right.

1 answer

To solve the inequality \(-\frac{1}{2}x \geq 4\), we can first isolate \(x\):

  1. Multiply both sides of the inequality by -2. Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign:

\[ x \leq -8 \]

This means that the solution set includes all numbers less than or equal to -8.

Now, looking at the options provided:

  1. A number line from negative 10 to 10 in increments of 2, with a bold line starting at -2 and pointing to the left, represents \(x < -2\).
  2. A number line from negative 10 to 10 in increments of 2, with a bold line starting at -8 and pointing to the left, represents \(x < -8\).
  3. A number line from negative 10 to 10 in increments of 2, with a bold line starting at -2 and pointing to the right, represents \(x > -2\).
  4. A number line from negative 10 to 10 in increments of 2, with a bold line starting at -8 and pointing to the right, represents \(x > -8\).

The correct representation of the solution \(x \leq -8\) is the second option: a bold line starting at -8 and pointing to the left.

So the answer is:

A number line from negative 10 to 10 in increments of 2. A point is at negative 8 and a bold line starts at negative 8 and is pointing to the left.