To solve the equation [tex]\( |x - 7| = 4 \)[/tex], we need to consider the definition of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, if [tex]\( |x - 7| = 4 \)[/tex], [tex]\( x - 7 \)[/tex] could be either 4 or -4.
We can break this down into two separate cases:
Case 1:
[tex]\[ x - 7 = 4 \][/tex]
To find [tex]\( x \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ x - 7 = 4 \][/tex]
[tex]\[ x = 4 + 7 \][/tex]
[tex]\[ x = 11 \][/tex]
Case 2:
[tex]\[ x - 7 = -4 \][/tex]
Again, solve for [tex]\( x \)[/tex]:
[tex]\[ x - 7 = -4 \][/tex]
[tex]\[ x = -4 + 7 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the solutions to the equation [tex]\( |x - 7| = 4 \)[/tex] are:
[tex]\[ x = 11 \text{ or } x = 3 \][/tex]
So, the final answers are:
[tex]\[ 11, 3 \][/tex]
