Question

suppose that p and q are points on the number line. if pq = 5 and q lies at -12, where could p be located? if there is more than one location, separate them with commas. location(s) of p :

Explanation:

Step1: Recall distance on number line

The distance between two points \( P \) (with coordinate \( x \)) and \( Q \) (with coordinate \( -12 \)) on the number line is given by \( |x - (-12)|=|x + 12| \). We know this distance \( PQ = 5 \), so \( |x+12| = 5 \).

Step2: Solve the absolute - value equation

The absolute - value equation \( |a|=b \) (where \( b\geq0 \)) has two solutions: \( a = b \) or \( a=-b \).
For \( |x + 12|=5 \), we have two cases:

• Case 1: \( x+12 = 5 \)

Subtract 12 from both sides of the equation: \( x=5 - 12=-7 \).

• Case 2: \( x + 12=-5 \)

Subtract 12 from both sides of the equation: \( x=-5 - 12=-17 \).

Answer:

-7, -17