Solving Linear Inequalities

1. Represent solutions to inequalities graphically and using set notation.

Graphing InequalitiesGraphing Inequalities

Parentheses/bracket method :Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥

x < 2

x ≥ 2

Open Circle/closed circle method:Open Circle: endpoint is not included <, > Closed Circle: endpoint is included ≤, ≥

x < 2

x ≥ 2

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

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

InequalitiesInequalities

[

Graph, then write interval notation and set-builder notation.

x ≥ 5

Interval Notation: [ 5, ∞)

x < –3

Interval Notation: (– ∞, –3)

)

InequalitiesInequalitiesGraph, then write interval notation and set-builder notation.

1 < a < 6

Interval Notation: ( 1, 6 )

–7 < x ≤ 3

Interval Notation: (– 7, –3]

( )

( ]

Rewrite in interval notation and graph

X ≤ 5

ORDER and INTERVAL NOTATIONGRAPHING THE NUMBER SETS ON THE NUMBER LINE

GRAPH: -1 < x < 3

0 2 4 6-2-4-6

INTERVAL NOTATION (AND TYPE)

(-1, 3 ) Bounded & Open

ORDER and INTERVAL NOTATIONGRAPHING THE NUMBER SETS ON THE NUMBER LINE

GRAPH: -6 < x < -½

0 2 4 6-2-4-6

INTERVAL NOTATION (AND TYPE)

[-6, -½ ] Bounded & Closed

ORDER and INTERVAL NOTATIONGRAPHING THE NUMBER SETS ON THE NUMBER LINE

GRAPH: 0 < x < 2.3

0 2 4 6-2-4-6

INTERVAL NOTATION (AND TYPE)

( 0, 2.3 ] Bounded & Half-Open

ORDER and INTERVAL NOTATIONGRAPHING THE NUMBER SETS ON THE NUMBER LINE

GRAPH: x > -2

0 2 4 6-2-4-6

INTERVAL NOTATION (AND TYPE)

(-2, ∞) Unbounded &Open

ORDER and INTERVAL NOTATIONGRAPHING THE NUMBER SETS ON THE NUMBER LINE

GRAPH: x < 4.5

0 2 4 6-2-4-6

INTERVAL NOTATION (AND TYPE)

(- ∞, 4.5 ] Unbounded &Closed

ORDER and INTERVAL NOTATIONConverting between intervals and inequalities

Write (-2, 5) as an inequalityin set-builder notation

{x | -2 < x < 5}Open or Closed?

Open

ORDER and INTERVAL NOTATION

Write [-2, ∞) as an inequalityin set-builder notation

{x | x > -2}Open or Closed?

Closed

Converting between intervals and inequalities

ORDER and INTERVAL NOTATION

Write (-∞, 6) as an inequalityin set-builder notation

{x | x < 6}Open or Closed?

Open

Converting between intervals and inequalities