Download Solving a Linear Inequality

Solving a Linear Inequality
Solving an Inequality
In order to find the points that satisfy an
inequality statement:
1.
Find the boundary
2. Test every region to find which one(s)
satisfies the original statement
Finding an Inequality Boundary
Boundary Point: A solution(s) that makes the
inequality true (equal). It could be the smallest
number(s) that make it true. Or it is the largest
number(s) that makes it NOT true.
EX: Find the boundary point of 2x  5  3
To find a boundary replace the
inequality symbol with an equality
symbol.
2x  5  3
2x  8
x4
Solving a 1 Variable Inequality
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
3  2x  1
Find the Boundary
Test Every Region
Graphical
Solution
x
Change inequality to equality
Solve
3  2x  1
2 x  2
x 1
Plot Boundary Point(s)
0
Pick a point in
each region
x=0
Substitute
into Original
Shade True
Region(s)
x=2
3  2  0  1 3  2  2  1
3<1
False
x 1
-1 < 1
True
Algebraic
Solution
Solving a 1 Variable Inequality: The Answer
is All Numbers
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
9k  4 1  2k  3  7k
Find the Boundary
Graphical
Solution
Test Every Region
x
Change inequality to equality
9k  4 1  2k  3  7k
0
Solve
9k  3  9k  3
00
Since every value of k satisfies the
equation, every Point is a Boundary Point
All Numbers
“Algebraic”
Solution
Solving a 1 Variable Inequality: No Solutions
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
3y 1  3y 1
Find the Boundary
Graphical
Solution
Test Every Region
x
Change inequality to equality
3y 1  3y 1
Solve
0
00
Since every value of k satisfies the
equation, every Point is a Boundary Point
No Solution
“Algebraic”
Solution
Solving an Absolute Value Inequality
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
x2 3
Find the Boundary
Test Every Region
Graphical
Solution
x
Change inequality to equality
Solve
x2 3
x  2  3
x  2  3 x  2  3
x  5 or x  1
Plot Boundary Point(s)
Pick a point in
each region
x = -2
2  2  3
Substitute
into Original
4>3
True
Shade True
Region(s)
0
x=0
02 3
2>3
False
x=6
62 3
4>3
True
x  1 or x  5
Algebraic
Solution
Solving a Quadratic Inequality
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
x  2x 1  4
2
Find the Boundary
Graphical
Solution
Test Every Region
x
Change inequality to equality
x  2x 1  4
x=2
x=0
x = -4
2
x  2 x  3  0 Substitute
 4   2  4   1  4
 2  2  2  1  4
 0  2  0  1  4
into
Original
 x  3 x 1  0
1<4
9≤4
9<4
2
Solve
0
Pick a point in
each region
2
2
2
x  3 or x  1
Plot Boundary Point(s)
False
Shade True
Region(s)
True
3  x  1
False
Algebraic
Solution
Solving a 1 Variable Inequality
Represent the solutions to the following inequality
algebraically and on a number line.
Closed or Open Dot(s)?
2 x  5x  3  x  4 x  3
2
2
Find the Boundary
Graphical
Solution
Test Every Region
x
Change inequality to equality
2 x2  5x  3  x2  4 x  3
Solve
x  x  6  0 Substitute
 x  3 x  2  0 into Original 9 ≤ 3
2
x  3 or x  2
Plot Boundary Point(s)
0
Pick a point in
each region
x = -4
x=3
x=0
2  4   5  4   3   4   4  4   3
2  3  5  3  3   3  4  3   3
2
2
2  0  5  0  3  0  4 0  3
2
2
False
Shade True
Region(s)
2
-3 ≤ 3
True
2
30 ≤ 24
False
3  x  2
Algebraic
Solution