Algebra 1 Common Core Student Edition, Grade 8-9

What must be true
of the expression
100 - 20f?
Its a b solute value is 40,
so 100 - 201 must
equal either 40 or - 4 0.
Use th is fa c t to w rite and
solve tw o equations.

How can you make
the equation look
like one you've
solved before?
Use p ro perties o f e q u a lity
to isolate the absolute
value expression on one
side o f th e equal sign.

Some equations, such as 12x - 5 1 = 13, have variable expressions w ithin absolute

value symbols. The equation \2x - 5| = 13 m ean s th a t th e distance on a n u m b e r line

from 2x - 5 to 0 is 13 units. There are two points th at are 13 units from 0:13 a n d - 1 3.

So to find the values of x, solve the equations 2x - 5 = 13 and 2x - 5 = - 1 3. You can

generalize this process as follows.

Key Concept Solving Absolute Value Equations

To solve a n e q u a tio n in t h e fo rm | A | = b, where A rep resen ts a variable expression

and b > 0, solve A = b and A = —b.

v _ _ J

Solving an Absolute Value Equation

Multiple Choice Starting from 100 ft away, your friend skates toward you and then

passes by you. She skates at a constant speed of 20 ft/s. Her distance d from you in

feet after t seconds is given by d = 11 0 0 — 2011. At what times is she 40 ft from you?

CS> —2 s a n d 8 s CID _ 3 s a n d 7 s CC^> 3 s an d 7 s Q D 2 s an d 8 s

100 - 201 = 4 0 «— Write two equations. —* 100 - 20 1 = - 4 0

-201= -60 «— Subtract 100 from each side. —> - 2 0 1 = - 1 4 0

1 = 3 <— Divide each side by —20. —> 1 = 7

The solutions are 3 s an d 7 s. The correct answ er is C.

0 G ot It? 2. A nother friend’s distance d from you (in feet) after 1 seconds is given by

d = 180 - 511. W hat does th e 80 in th e eq u a tio n represent? W hat does the

5 in th e equation represent? At w hat tim es is she 60 ft from you?

Recall th a t absolute value represents distance from 0 o n a n u m b e r line. D istance is

always nonnegative. So any equ atio n th a t states th a t the absolute value of an expression

is negative has no solutions.

Pr o b l em 3 Solving an Absolute Value Equation W ith No Solution

What are the solutions of 312z + 9| + 12 = 10?

312z + 91 + 12 = 10

3|2z + 9 1 = - 2 Subtract 12 from each side.

|2z + 9|=-| Divide each side by 3.

The absolute value of an expression can n o t be negative, so there is no solution.

0 Got It? 3. W hat are th e solutions of|3x-6|-5 = -7?

208 Ch a p t e r 3 So l v i n g I n e q u a l i t i e s