Exercise 50 Page 259

An absolute value measures an expression's distance from a midpoint on a number line.

6 = ∣ a - 7 ∣ \Leftrightarrow ∣ a - 7 ∣ = 6

This equation means that the distance is

6,

either in the

positive

direction or the

negative

direction.

∣ a - 7 ∣ = 6 \Rightarrow a - 7 = - 6 a - 7 = - 6

To find the solutions to the absolute value equation, we need to solve both of these cases for

a .

∣ a - 7 ∣ = 6

$a - 7 \geq 0 : a - 7 = 6 a - 7 < 0 : a - 7 = - 6 (I) (II)$

a - 7 = - 6 a - 7 = - 6 (I) (II)

$(I), (II):$ $LHS + 7 = RHS + 7$

a_{1} = 13 a_{2} = 1

Both

13

and

1

are solutions to the absolute value equation.