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. (-∞,∞)