Question

graph the solution to the inequality on the number line.
\\(|v - 7| \geq 3\\)

Explanation:

Step1: Solve the absolute - value inequality

The absolute - value inequality \(|v - 7|\geq3\) can be split into two separate inequalities according to the definition of absolute value. The definition of absolute value states that if \(|x|\geq a\) (\(a>0\)), then \(x\geq a\) or \(x\leq - a\).
For \(|v - 7|\geq3\), we have two cases:
Case 1: \(v - 7\geq3\)
Add 7 to both sides of the inequality: \(v-7 + 7\geq3 + 7\), which simplifies to \(v\geq10\).
Case 2: \(v - 7\leq - 3\)
Add 7 to both sides of the inequality: \(v-7 + 7\leq - 3+7\), which simplifies to \(v\leq4\).

Step2: Graph the solution on the number line

• For the inequality \(v\geq10\): We use a closed circle (because the inequality is "greater than or equal to") at \(v = 10\) and draw an arrow to the right (since \(v\) is greater than or equal to 10).
• For the inequality \(v\leq4\): We use a closed circle (because the inequality is "less than or equal to") at \(v = 4\) and draw an arrow to the left (since \(v\) is less than or equal to 4).

Answer:

To graph the solution of \(|v - 7|\geq3\) (which is \(v\leq4\) or \(v\geq10\)) on the number line:

• Place a closed circle at \(4\) and draw an arrow to the left (towards negative infinity) from \(4\).
• Place a closed circle at \(10\) and draw an arrow to the right (towards positive infinity) from \(10\).