warm-up – pick up handout up front solve by factoring. 1000x 3 -10x answers: 1.x=0, x=1/10, x=...
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Warm-up – pick up handout up front
Solve by factoring. 1000x3-10x
Answers:
1.x=0, x=1/10, x= -1/10
HW 1.7A (2-14 evens, 21-24, 39-47)
3 2 4 4x
2.
1.
2.
2
3
Solve for x
HW 1.6B (31-39 all)HW 1.6C (61-75 odds)
-1 34. 13 33. 932.x 4.31 xxx
3 38. 68 37. 2736.x 4.35 55 xxx45.39 andx
23 65. 59 63. 88.61 andxandxandx
45
4 71.
5
2
5
2 69.
3
53.67 andxandxandx
2
1 75.solution no .73 x
Lesson 1.7A Solving linear inequalities and the types of notation
Objective: To be able to use interval notation when solving linear inequalities, recognize inequalities with no solution or all real numbers as a solution.
The set of all solutions is called the solution set of the inequality. Set-builder notation and a new notation called interval notation, are used to represent solution sets. (See Handout!)
Interval Notation: See Handout
Example 1: (-1,4] Graph and write the solution in set-builder notation.
You Try: (-∞, -4)
(-1,4] = {x -1 < x ≤ 4}
)
-4 0
( ]
-1 0 4
(-∞, -4) = { x x < -4}
Interval Notation Set Builder Notation
The answer is read x such that -1 is less than x which is less than or equal to 4.
Graphing intervals on a number line!Example 2:
Graph each interval on a number line.
{x 2 < x < 3}
( ]
2 3
(2,3]
Write answer using set builder notation.
You try: Graph and write in set builder notation.
{x 1 < x < 6}
(-3,7] {x -3 < x < 7}
( ]
-3 7
[ )
1 6
[1,6)
Example 3: Solve and graph this linear inequality.
-2x - 4 > x + 5
x < -3
)
-3 0
Remember to switch the sign when you multiply or divide by a negative.
You Try!!
3x+1 > 7x – 15
Use interval notation to express the solution set.
Graph the solution.
Answer: (-∞, 4) )
4
Inequalities with Unusual Solution Sets
Some inequalities have no solution.
Example 4: x > x+1
There is no number that is greater than itself plus one.
The solution set is an empty set ( this is a zero with a slash through it)
Notes Continued:Like wise some inequalities are true for all real
numbers such as:
Example 5: x<x+1.
Every real number is less than itself plus 1.
The solution set is { x x is a real number}
Interval notation, or all real numbers. ),(
Notes Continued:When solving an inequality with no solution,
the variable is eliminated and there will be a false solution such as 0 > 1.
When solving an inequality that is all real numbers, the variable is eliminated and there will be a true solution such as 0 < 1.
Example 6:Solving a linear inequality for a solution set.
A. 2 (x + 4) > 2x + 3
B. x + 7 < x – 2
The inequality 8>3 is true for all values of x. The solution set is
{x x is a real number} or ,
solution: 8 > 3 solution: 7 < -2
The inequality 7 < -2 is false for all values of x. The solution set is
You try: 3(x + 1) > 3x + 2
3x + 3 > 3x + 2
3 > 2
Summary: Describe the ways in which solving a linear inequality is different from solving a linear equation.
The solution set is all real numbers. (-∞,∞)