Inequalities and Interval Notation Section 1.5 Section 1.5 Linear Inequalities and Interval Notation Inequalities and Interval Notation

±∞ always gets a parentheses Interval Notation ) or ( means “not equal to” or not inclusive. In middle school these would have been “open” circle markers from < or > symbols. ] or [ means “equal to” or inclusive. These would have been “closed” circle markers from symbols. ±∞ always gets a parentheses Written with smallest desired number on left, largest desired number on the right.

) EXAMPLE Graph simple inequalities Many times instead of using inequality symbols we will use a new notation called Interval Notation… Graph x < 2. The solutions are all real numbers less than 2. A parenthesis is used in the graph to indicate 2 is not a solution. ) Instead of using open/closed dots, we will now use parenthesis and brackets to indicate exclusive/inclusive. Just like interval notation.

[ EXAMPLE Graph simple inequalities Interval Notation: Graph x ≥ –1. The solutions are all real numbers greater than or equal to –1. A bracket is used in the graph to indicate –1 is a solution. [

( ) EXAMPLE Graph compound inequalities Graph –1 < x < 2. Interval Notation: The solutions are all real numbers that are greater than –1 and less than 2. ( )

( ] EXAMPLE Graph compound inequalities Graph x ≤ –2 or x > 1. Interval Notation: (-∞, -2] U (1, ∞) The U means “union”…the useful values can come from either interval. Many times we take it to mean “or” The solutions are all real numbers that are less than or equal to –2 or greater than 1. ( ]

Graphing Compound Inequalities Rewrite the interval as a single interval if possible. (-∞, 5)∩(-2, ∞) The intersection symbol ∩ means “and”. This desired result has to satisfy BOTH intervals.

Graph [-4,5)∩[-2,7)

Graph (-7, 3)U[0, 5)

Graph (-∞, -2]U[-2, ∞)

Rewrite in interval notation and graph X ≤ 5

Write the inequality in interval notation

Solve an inequality with a variable on one side EXAMPLE Solve an inequality with a variable on one side 20 + 1.5g ≤ 50. 20 + 1.5g ≤ 50 Write inequality. 1.5g ≤ 30 Subtract 20 from each side. g ≤ 20 Divide each side by 1.5. ANSWER (-∞, 20 ]

) EXAMPLE Solve an inequality with a variable on both sides Solve 5x + 2 > 7x – 4. Then graph the solution. 5x + 2 > 7x – 4 Flip the inequality when multiplying or dividing both sides by a negative #. – 2x + 2 > – 4 – 2x > – 6 x < 3 ANSWER The solutions are all real numbers less than 3. The graph is shown below. ) (-∞, 3)

Solve the inequality and express in interval notation

Solve and write the answer in interval notation

You rent a car for two days every weekend for a month You rent a car for two days every weekend for a month. They charge you $50 per day, as well as $.10 per mile. Your bill has ranged everywhere from $135 to $152. What is the range of miles you have traveled?

GUIDED PRACTICE Solve the inequality. Then graph the solution. 4x + 9 < 25 5x – 7 ≤ 6x x < 4 (-∞, 4) ANSWER x > – 7 [-7,∞) ANSWER 3 – x > x – 9 1 – 3x ≥ –14 x ≤ 5 (-∞, 5] ANSWER x < 6 (-∞, 6) ANSWER