3. Absolute Value Equations
⚫ You should recall that the absolute value of a number a,
written |a|, gives the distance from a to 0 on a number
line.
⚫ By this definition, the equation |x| = 3 can be solved by
finding all real numbers at a distance of 3 units from 0.
Both of the numbers 3 and ‒3 satisfy this equation, so
the solution set is {‒3, 3}.
4. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
=x a
− =9 4 7x
5. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
The solution set is
=x a
− =9 4 7x
− =9 4 7x − = −9 4 7x
− = −4 2x − = −4 16x
=
1
2
x = 4x
or
1
,4
2
6. Absolute Value Inequalities
⚫ For absolute value inequalities, we make use of the
following two properties:
⚫ |a| < b if and only if –b < a < b.
⚫ |a| > b if and only if a < –b or a > b.
⚫ Example: Solve − + 5 8 6 14x
7. Absolute Value Inequalities (cont.)
⚫ Example: Solve
The solution set is
− + 5 8 6 14x
or
− 5 8 8x
− −5 8 8x − 5 8 8x
− −8 13x − 8 3x
13
8
x −
3
8
x
− −
3 13
, ,
8 8
8. Special Cases
⚫ Since an absolute value expression is always
nonnegative:
⚫ Expressions such as |2 – 5x| > –4 are always true. Its
solution set includes all real numbers, that is, (–, ).
⚫ Expressions such as |4x – 7| < –3 are always false—
that is, it has no solution.
⚫ The absolute value of 0 is equal to 0, so you can solve
it as a regular equation.
9. Classwork
⚫ 1.8 Assignment (College Algebra)
⚫ Page 163: 10-22 (even), page 155: 36-52 (4),
page 144: 72-88 (4)
⚫ 1.8 Classwork Check
⚫ Quiz 1.7
Your six weeks test is next Friday.
