Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. x ≤ 7 23 < –2x + 3 Solve each inequality and graph the solutions. 4. 4x + 1 ≤ 25 x ≤ 6 5. 0 ≥ 3x + 3 –1 ≥ x

Solving Compound Inequalities

Objectives Vocabulary compound inequality intersection union Solve compound inequalities with one variable. Graph solution sets of compound inequalities with one variable. Vocabulary compound inequality intersection union

The inequalities you have seen so far are simple inequalities The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality. Copy

CONJUNCTION Two or more sentences are linked by the word and. Ex. - 2< x and x <1 which can be written - 2 < x < 1 . Ex. 5 > x and x > 3 which can be written 5 > x >3 . Ex. x < - 2 and x < 1 which can be written x < - 2 . CONJUNCTION

How do I solve conjunctions? 4x + 5 > -7 AND 4x + 5 ≤ 25 First rewrite using a single inequality -7 < 4x + 5 ≤ 25 Next you need to isolate the variable following the steps of algebra. Ask yourself: Can I do distributive property or combine like terms in the center section? If so do this step, if not move on! How do I solve conjunctions?

What now? Isolate the variable in the middle. By adding or subtracting the number term on each side of both symbols (middle, left, and right). -7 < 4x + 5 ≤ 25 -5 -5 -5 -12< 4x ≤ 20 What now?

Multiply or divide by the coefficient on each side of both symbols (middle, left, and right). Remember if you divide or multiply by a negative number you MUST reverse the inequality signs! -12< 4x ≤ 20 4 4 4 -3< x ≤ 5 Next step?

Final Step Check the answer for reasonableness Graph the solution -5 0 5 Final Step

Solving Compound Inequalities Involving AND Solve the compound inequality and graph the solutions. Copy 8 < 3x – 1 ≤ 11 8 < 3x – 1 ≤ 11 +1 +1 +1 9 < 3x ≤ 12 Since 1 is subtracted from 3x, add 1 to each part of the inequality. Since x is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication. 3 < x ≤ 4

Graph the intersection by finding where the two graphs overlap. Continued Graph 3 < x. Graph x ≤ 4. Graph the intersection by finding where the two graphs overlap. –5 –4 –3 –2 –1 1 2 3 4 5

Disjunction Two or more sentences are linked by the word or . Ex. - 2 < x or x > 1 Ex. x < - 2 or x < 1 which can be written x < 1 . Ex. x > - 2 or x < 1 which is the same as -2 < x <1 Disjunction

So how do I solve disjunctions? 8 + 2x < 6  OR  3x - 2 > 13 Solve each inequality. The solution must be written with two inequalities connected with “OR”. 3x - 2 > 13 +2 +2 3x > 15 3 3 x > 5 8 + 2x < 6 -8 -8 2x < -2 2 2 x < -1 OR So how do I solve disjunctions?

What’s next? Graph each inequality. One of the circles goes on each number in the solution. The darkened bar is graphed in the direction indicated by the symbol with the number. If the darkened bars are going toward each other, the answer is All Real Numbers, so you would graph a darkened bar over the entire number line. If both darkened bars are going to the right, the answer is all number > or ≥ the smallest value. If both darkened bars are going to the left, the answer is all number < or ≤ the largest value. -5 -1 0 5 What’s next?

Solving Compound Inequalities Involving OR Solve the inequality and graph the solutions. 4x ≤ 20 OR 3x > 21 Copy 4x ≤ 20 OR 3x > 21 x ≤ 5 OR x > 7 Solve each simple inequality. Graph x ≤ 5. Graph x > 7. Graph the union by combining the regions. –10 –8 –6 –4 –2 2 4 6 8 10

Every solution of a compound inequality involving AND must be a solution of both parts of the compound inequality. If no numbers are solutions of both simple inequalities, then the compound inequality has no solutions. The solutions of a compound inequality involving OR are not always two separate sets of numbers. There may be numbers that are solutions of both parts of the compound inequality.

Lesson Quiz: Part I 1. The target heart rate during exercise for a 15 year-old is between 154 and 174 beats per minute inclusive. Write a compound inequality to show the heart rates that are within the target range. Graph the solutions. 154 ≤ h ≤ 174 Solve each compound inequality and graph the solutions. 2. 2 ≤ 2w + 4 ≤ 12 –1 ≤ w ≤ 4 3. 3 + r > −2 OR 3 + r < −7 r > –5 OR r < –10

Lesson Quiz: Part II Write the compound inequality shown by each graph. 4. x < −7 OR x ≥ 0 5. −2 ≤ a < 4