What is the solution to |x − 6] ≥ 5?
The Correct Answer and Explanation is:
The inequality given is ( |x – 6| \geq 5 ).
Step 1: Understanding the absolute value inequality
The absolute value inequality ( |A| \geq B ) means that the quantity inside the absolute value (denoted as ( A )) is either greater than or equal to ( B ), or less than or equal to ( -B ). In other words, if ( |A| \geq B ), then:
[
A \geq B \quad \text{or} \quad A \leq -B
]
Thus, for our inequality ( |x – 6| \geq 5 ), this translates to:
[
x – 6 \geq 5 \quad \text{or} \quad x – 6 \leq -5
]
Step 2: Solve each inequality separately
Case 1: ( x – 6 \geq 5 )
Solve for ( x ) by adding 6 to both sides:
[
x \geq 5 + 6
]
[
x \geq 11
]
Case 2: ( x – 6 \leq -5 )
Solve for ( x ) by adding 6 to both sides:
[
x \leq -5 + 6
]
[
x \leq 1
]
Step 3: Combine the solutions
The solution to the inequality ( |x – 6| \geq 5 ) is the union of the two inequalities:
[
x \geq 11 \quad \text{or} \quad x \leq 1
]
Step 4: Final solution
Thus, the solution to ( |x – 6| \geq 5 ) is:
[
x \leq 1 \quad \text{or} \quad x \geq 11
]
In interval notation, this is written as:
[
(-\infty, 1] \cup [11, \infty)
]
Explanation
The absolute value function measures the distance of a number from zero, regardless of direction. In this inequality, ( |x – 6| \geq 5 ) indicates that the distance between ( x ) and 6 must be at least 5 units. This means ( x ) must either be 5 or more units to the right of 6 (i.e., ( x \geq 11 )) or 5 or more units to the left of 6 (i.e., ( x \leq 1 )). Therefore, the solution consists of all values of ( x ) in these two ranges, which are ( x \leq 1 ) or ( x \geq 11 ).