Compound Inequalities A compound inequality is either two inequalities separated by a word, or an expression in between two inequality symbols.

Compound Inequalities A compound inequality is either two inequalities separated by a word, or an expression in between two inequality symbols. Let’s look at the first type…separated by a word

Compound Inequalities There are two words that can appear in between inequalities AND / OR

Compound Inequalities There are two words that can appear in between inequalities AND / OR With AND, the solution set is what is shared or common

Compound Inequalities There are two words that can appear in between inequalities AND / OR With AND, the solution set is what is shared or common With OR, the solution set is all solutions combined

Compound Inequalities There are two words that can appear in between inequalities AND / OR With AND, the solution set is what is shared or common With OR, the solution set is all solutions combined We are going to use a line graph to help us find the solution set…

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 2𝑥+5>−15

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 +2=+2 3𝑥≤12 3𝑥 3 ≤ 12 3 𝑥≤4 2𝑥+5>−15 −5=−5 2𝑥>−20 2𝑥 2 > −20 2 𝑥>−10

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points - start with 𝑥≤4, it’s a closed circle with the arrow going left -10 4

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points - start with 𝑥≤4, it’s a closed circle with the arrow going left - now graph 𝑥>−10. open circle with arrow pointing right -10 4

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points STEP # 3 : With AND in between the expressions, we need to find where the graphs are ON TOP of each other -10 4

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points STEP # 3 : With AND in between the expressions, we need to find where the graphs are ON TOP of each other THIS IS OUR SHARED AREA -10 4

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points STEP # 3 : With AND in between the expressions, we need to find where the graphs are ON TOP of each other Mesh the ”shared area “ into one line -10 4

Compound Inequalities EXAMPLE #1 : Find the solution for 3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15 STEP # 1 : Solve each inequality 3𝑥−2≤10 𝑥≤4 2𝑥+5>−15 𝑥>−10 STEP # 2 : Set up a number line and graph your points STEP # 3 : With AND in between the expressions, we need to find where the graphs are ON TOP of each other Mesh the ”shared area “ into one line -10 4 Solution is −10<𝑥≤4

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11 STEP #1 : Solve each inequality 6−𝑥≤1 −6 =−6 −𝑥≤−5 −𝑥 −1 ≤ −5 −1 𝑥≥5 4𝑥−1>11 +1=+1 4𝑥>12 4𝑥 4 > 12 4 𝑥>3 Remember, when you divide by a negative, the inequality changes direction

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11 STEP #1 : Solve each inequality 6−𝑥≤1 𝑥≥5 4𝑥−1>11 𝑥>3 STEP #2 : Set up a number line, graph your points and draw your arrows 3 5

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11 STEP #1 : Solve each inequality 6−𝑥≤1 𝑥≥5 4𝑥−1>11 𝑥>−5 closed circle and to the right open circle and to the right STEP #2 : Set up a number line, graph your points and draw your arrows 3 5

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11 STEP #1 : Solve each inequality 6−𝑥≤1 𝑥≥5 4𝑥−1>11 𝑥>−5 closed circle and to the right open circle and to the right STEP #2 : Set up a number line, graph your points and draw your arrows FINAL graph 3 5 STEP # 3 : With OR in between the expressions, we will look at all possible solutions. Since the arrow for (3) joins and is in the same direction as (+5), the solution set is all numbers greater than (3)

Compound Inequalities EXAMPLE #2 : Find the solution for 6−𝑥≤1 𝑂𝑅 4𝑥−1>11 STEP #1 : Solve each inequality 6−𝑥≤1 𝑥≥5 4𝑥−1>11 𝑥>−5 closed circle and to the right open circle and to the right STEP #2 : Set up a number line, graph your points and draw your arrows FINAL graph 3 5 Solution is 𝑥>3

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 3𝑥<21 If I covered the left side of the equation and only solved for the right side, I would add 8 to both sides…

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 = +8 9<3𝑥<21 When solving these types, whatever you would do to solve one side, you would also do to solve the other side…

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 = +8 9<3𝑥<21 9 3 < 3𝑥 3 < 21 3 Now divide EVERYTHING by the coefficient of 𝑥 which is 3

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 = +8 9<3𝑥<21 9 3 < 3𝑥 3 < 21 3 3<𝑥<7

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 = +8 9<3𝑥<21 3 7 9 3 < 3𝑥 3 < 21 3 Set up your number line and graph the points… 3<𝑥<7 Open circles

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 Solution is 3<𝑥<7 +8 = +8 = +8 9<3𝑥<21 3 7 9 3 < 3𝑥 3 < 21 3 If both inequality symbols point to the LEFT, the solution is SQUEEZED in between your points… 3<𝑥<7 Open circles

Compound Inequalities The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities. EXAMPLE # 3 : 1<3𝑥−8<13 +8 = +8 = +8 9<3𝑥<21 3 7 9 3 < 3𝑥 3 < 21 3 If both inequality symbols point to the LEFT, the solution is SQUEEZED in between your points… 3<𝑥<7 Open circles 3 7 If both inequality symbols point to the RIGHT, the solution HAS A GAP in between your points… Solution is 3>𝑥>7

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3 With this in mind, we solve absolute value equations like this… EXAMPLE : Find the solution set for 𝑥+3 <10

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3 With this in mind, we solve absolute value equations like this… EXAMPLE : Find the solution set for 𝑥+3 <10 −10<𝑥+3<10 Set up a compound inequality with the negative of the original answer on the left side. Keep the inequality symbol pointed in the same direction. Solve like we did with compound inequalities.

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3 With this in mind, we solve absolute value equations like this… EXAMPLE : Find the solution set for 𝑥+3 <10 −10<𝑥+3<10 −3 = −3=−3 Subtract 3

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3 With this in mind, we solve absolute value equations like this… EXAMPLE : Find the solution set for 𝑥+3 <10 −10<𝑥+3<10 −3 = −3=−3 −13<𝑥<7 Subtract 3

Compound Inequalities ABSOLUTE VALUE INEQUALITIES Recall that all absolute values are positive. So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3 With this in mind, we solve absolute value equations like this… EXAMPLE : Find the solution set for 𝑥+3 <10 −10<𝑥+3<10 −3 = −3=−3 −13<𝑥<7 ** If we would graph this, both inequalities point left so a “squeeze” Subtract 3 −13 7