To solve the inequality [tex]\( |2x - 6| > 14 \)[/tex], we need to consider the definition of absolute value and split the inequality into two separate cases. The absolute value [tex]\( |A| \)[/tex] of an expression [tex]\( A \)[/tex] is defined as:
[tex]\[ |A| = A \quad \text{if} \quad A \geq 0 \][/tex]
[tex]\[ |A| = -A \quad \text{if} \quad A < 0 \][/tex]
Given [tex]\( |2x - 6| > 14 \)[/tex], this means:
[tex]\[ 2x - 6 > 14 \quad \text{or} \quad 2x - 6 < -14 \][/tex]
We will solve these two inequalities separately:
### Case 1: [tex]\( 2x - 6 > 14 \)[/tex]
1. Add 6 to both sides:
[tex]\[ 2x > 20 \][/tex]
2. Divide both sides by 2:
[tex]\[ x > 10 \][/tex]
### Case 2: [tex]\( 2x - 6 < -14 \)[/tex]
1. Add 6 to both sides:
[tex]\[ 2x < -8 \][/tex]
2. Divide both sides by 2:
[tex]\[ x < -4 \][/tex]
Now we combine the solutions from both cases. Thus, the solution to the inequality [tex]\( |2x - 6| > 14 \)[/tex] is:
[tex]\[ x < -4 \quad \text{or} \quad x > 10 \][/tex]
In interval notation, this can be written as:
[tex]\[ (-\infty, -4) \cup (10, \infty) \][/tex]
This solution indicates that [tex]\( x \)[/tex] must be either less than [tex]\(-4\)[/tex] or greater than [tex]\(10\)[/tex].
