Module 6: Inequalities And Absolute Values

Question 1

Graphing inequalities on a number line

Answer 1

Use a closed circle to indicate that the threshold is a possible quantity of X IF the inequality is inclusive. If exclusive, leave the circle open to indicate the opposite

Question 2

Important rule when multiplying or dividing inequalities

Answer 2

If the inequality is multiplied or divided by a negative number, the inequality sign must be reversed (from < to > or vice versa)

This rule doesn’t apply to addition or subtraction - those can be done normally

This means that we can’t simplify inequalities by multiplying or dividing by a variable unless we know if the variable is negative or positive.

Question 3

Separate inequalities can be added together if ____

Answer 3

the inequality signs are facing the same direction

(If 2 inequalities have 2 different signs, look for opportunities to flip one sign by dividing/multiplying by a negative)

Question 4

Working with compound inequalities: Rules (2)

Answer 4

1) When simplifying, must take the same action on all three parts of the compound inequality
2) When a compound inequality is multiplied or divided by a negative number, BOTH signs must be reversed

Question 5

When I see a question asking the size relationship of variables across multiple inequalities, I will

Answer 5

consider setting up a number line. It can be tricky to keep multiple relationships straight & accurate without one.

Question 6

If x^2 < b, then

Answer 6

-sqrt(b) < x < sqrt(b)

Question 7

If x^2 > b, then

Answer 7

x > sqrt(b) OR x < -sqrt(b)

Question 8

Solving Equations with Absolute Values

Answer 8

Solve the equation twice:

• once assuming the absolute value is positive (just remove the bars & solve), and
• once assuming the absolute value is negative (convert the bars to parentheses and distribute a minus sign to all terms within)

Question 9

Absolute value & PEMDAS

Answer 9

When simplifying equations, the absolute value is the LAST step - isolate the abs value bars first

Question 10

Adding absolute values: a rule

Answer 10

|a+b| <= |a| + |b| (always true)

Question 11

If |a + b| = |a| + |b|, then one of two things must be true:

Answer 11

1) one of the values is zero

2) the values have the same sign (+/-)

Question 12

Subtracting absolute values: a rule

Answer 12

|a - b| >= |a| - |b| (always true)

Question 13

If |a - b| = |a| - |b|, then one of two things must be true:

Answer 13

1) the SECOND quantity is zero
2) both values share the same sign and the FIRST quantity is greater than or equal to the absolute value of the SECOND quantity.

Question 14

If |x| < b, then

Answer 14

x < b AND x > -b

Question 15

If |x| > b, then

Answer 15

x > b AND x < -b

Question 16

When I see that the absolute value of an expression is equal to a negative number, then I will _____

Answer 16

recognize that there are no possible solutions to that equation, because absolute values are always positive.

Question 17

When I encounter an absolute value equation with a variable on both sides of the equation, I will ____

Answer 17

CHECK THE SOLUTION BEFORE MOVING ON. It is possible for “extraneous solutions,” where the process is followed to a T but one of the solutions doesn’t actually yield the correct result when plugged in.