Interval Notation; Solving InequalitiesA.8A.8

Students will be able to (swbat):1. Represent Solutions to Inequalities

Graphically, and by Using Set & Interval Notation.

2. Solve Linear, Combined, and Absolute Value Inequalities.

Inequalities – Interval NotationInequalities – Interval Notation[( smallest, largest )]

Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥Infinity: always uses a parenthesis

x < 2

x ≥ 2

( –∞, 2)

[2, ∞)

4 < x < 9

(4, 9)3-part inequality

If the variable is on the left, the arrow points the same direction as the inequality.

Inequalities – Set NotationInequalities – Set Notation{variable | condition }

x < 2

x ≥ 2

( –∞, 2)

[2, ∞)

4 < x < 9

(4, 9)

{ x | x 5}

The set of all x such that x is greater than or equal to 5.

pipe

{ x | }

{ x | x ≥ 2}

{ x | 4 < x < 9}

x < 2

Inequalities

Graph, then write in interval and set notation.

1 < a < 6

Interval Notation: (-7,3] Set Notation: {x }|

Solving Inequalities Solving Inequalities

If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

164 x44

4x

164 x44

4x

164 x44

4x

Solving Inequalities Solving Inequalities

19494 kk

Solve then graph the solution and write it in interval notation and set notation.

155 k

kk 44

55

3k

]

Interval Notation:

Set Notation:

(– ∞, –3 ]

{ k | k ≤ –3 }

1954 k44

Solving Combined Inequalities

Solve the inequality 4 ≤ 2x + 2 ≤ 10, graph your solution, and write your solution in both interval and set notation.

Inequalities Involving Absolute Value

│x│< a is equivalent to -a < x < a (-a,a)│x│≤ a is equivalent to -a ≤ x ≤ a [-a,a]

│x│> a is equivalent to x< -a or x>a (-∞,-a)U(a,∞)│x│≥ a is equivalent to x≤ -a or x≥a (-∞,-a]U[a,∞)

Solving an Absolute Value Inequality

Solve │2u + 5 │≤ 7, graph the solution set, and writethe solution in both set and interval notation.

Methacton Merchandise

The Methacton Merchandise salesperson is paid a commission of $10 plus 50% of the selling price in excess of the cost to make the merchandise (owner’s cost). The owner of Methacton Merchandise claims that her products sell for at least owner’s cost plus $10 and at most owner’s cost plus $60. For each sale made, over what range can the salesperson expect the commission to vary?

Methacton Merchandise

10 + .5(10) = $15.0010 + .5(60) = $40.00

15.00 ≤ x ≤ 40.00