Download (-2) +

Solving Inequalities
• To solve an inequality, use the same
procedure as solving an equation with one
exception. When multiplying or dividing
by a negative number, reverse the
direction of the inequality sign.
•
-3x < 6
divide both sides by -3
-3x/-3 > 6/-3
x > -2
Solutions….
You can have a range of answers……
-5 -4 -3 -2 -1
0
All real numbers less than 3
x< 3
1
2
3
4
5
Solutions continued…
-5
-4 -3
-2 -1
0
All real numbers greater than -2
x > -2
1
2
3
4
5
Solutions continued….
-5 -4 -3 -2 -1
0
1
2
3
All real numbers less than or equal to 2
x2
4
5
Solutions continued…
-5 -4 -3 -2 -1
0
1
2
3
4
5
All real numbers greater than or equal to -3
x  3
Did you notice,
Some of the dots were
solid and some were open?
x2
-5 -4 -3 -2 -1
0
1 2 3 4
5
x0
-5 -4 -3 -2 -1
0 1
2 3 4
Why do you think that is?
If the symbol is > or < then dot is open because it can not be equal.
If the symbol is  or  then the dot is solid, because it can be that point too.
Where is -1.5 on the number
line? Is it greater or less than 2?
-2
-25
-20
-15
-10
-5
0
5
• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.
10
15
20
25
Solve an Inequality
w+5<8
We will use the same steps that we did with
equations, if a number is added to the variable, we
add the opposite sign to both sides:
w + 5 + (-5) < 8 + (-5)
w+0<3
w<3
All numbers less than 3
are solutions to this
problem!
THE TRAP…..
When you multiply or divide each side of
an inequality by a negative number, you
must reverse the inequality symbol to
maintain a true statement.
Solving using Multiplication
Multiply each side by the same positive number.
1
(2) x  3 (2)
2
x6
Solving Using Division
Divide each side by the same positive number.
3x  9
3
3
x3
Solving by multiplication of a
negative #
Multiply each side by the same negative number and REVERSE the
inequality symbol.
(-1)
 x  4 (-1)
Multiply by (-1).
See the switch
x  4
Solving by dividing by a
negative #
Divide each side by the same negative number and reverse
the inequality symbol.
 2x  6
-2
-2
x  3
Solving Inequalities
• 3b - 2(b - 5) < 2(b + 4)
3b - 2b + 10 < 2b + 8
b + 10 < 2b + 8
-b + 10 < 8
-b < -2
b>2
0 1 2
More Examples
x - 2 > -2
x + (-2) + (2) > -2 + (2)
x+0>0
x>0
All numbers greater than 0 make this problem true!
More Examples
4+y≤1
4 + y + (-4) ≤ 1 + (-4)
y + 0 ≤ -3
y ≤ -3
All numbers from -3 down (including -3) make this
problem true!
Solving compound inequalities is
easy if . . .
. . . you remember that a compound
inequality is just two inequalities
put together.
 5  2x  3  9
 5  2x  3 2x  3  9
 5  2x  3  9
You can solve them both at the
same time:
 5  2x  3  9
3 3
3
 8  2x  6
2 2 2
4 x3
Write the inequality from the
graph:
-25
1:
2:
3:
-20
-15
-10
-5
0
Write variable:
boundaries:
signs:
5
10
15
 10  x  5
20
25
IsSolve
this what
the inequality:
you did?
 15  4 x  7  5
7
7 7
 8  4 x  12
4 4 4
2  x 3
You did
to reverse
. .remember
.Good
didn’tjob!
you?
the signs . . .
 15  4 x  7  5
7
7 7
 8  4 x  12
4 4 4
2  x 3
Solving Absolute Value Inequalities
• Solving absolute value inequalities is a
combination of solving absolute value equations
and inequalities.
• Rewrite the absolute value inequality.
• For the first equation, all you have to do is drop the
absolute value bars.
• For the second equation, you have to negate the right
side of the inequality and reverse the inequality sign.
Solve: |2x + 4| > 12
2x + 4 > 12
2x > 8
x>4
or
2x + 4 < -12
2x < -16
x < -8
or
x < -8 or x > 4
-8
0
4
Solve: 2|4 - x| < 10
|4 - x| < 5
4-x<5
-x<1
x > -1
and
and
4 - x > -5
- x > -9
x<9
-1 < x < 9
-1 0
9