Inequalities and Absolute Values

Question 1

Inequality and Negative Numbers

Answer 1

When an inequality is multiplied or divided by a negative number, the inequality sign must be reversed.

Never divide/multiply a variable over an inequality (unless you know the sign of the variable).

Question 2

Adding/Subtracting Inequality Expressions

Answer 2

When the signs face the same direction, inequalities can be added/substracted in the same way equations are.

Question 3

Simplifying Inequalities with x^2

Answer 3

x^2 > 64
|x| > 8
x < -8 and x > 8

Question 4

Two Absolute Values Equal Each Other

Answer 4

If two absolute values are equal, it must be true that the expressions within absolute value bars are either equal or opposite.

Question 5

|a+b| <= |a| + |b|

Answer 5

The absolute value of the sum of two numbers, a and b, will always be less than or equal to the sum of the absolute values of the two numbers.

|a+b| <= |a| + |b| - This is a mathematical fact

IF |a+b| = |a| + |b|, then a,b must have the same sign

Question 6

|a-b| >= |a| - |b|

Answer 6

The absolute value of the subtraction of two variables, a and b, is greater than or equal to the absolute value of “a” minus absolute value of “b”.

|a-b| >= |a| - |b| - This is a mathematical fact

IF |a-b| = |a| - |b|, then a,b must have the same sign AND |a| >= |b|

Question 7

Extraneous Solutions

Answer 7

Absolute Value Equals a Negative Number = NO SOLUTION

If a variable within an absolute value sign equals an expression with the same variable, check for extraneous solutions.

If we solve the equation again, x = -1 is an extraneous solution

x -1 | = 2x —> x = -1 and 1/3

Question 8

Quadratic and Absolute Values

Answer 8

If a quadratic is equal to an expression with an absolute value, Isolate the absolute value expression first before factoring the quadratic.

Check for extraneous solutions if a quadratic expression equals an absolute value.

Question 9

Dividing variables over inequalities

Answer 9

You can divide over variables across the inequality if the stem tells you they are positive.

Question 10

Number Lines & Ranges

Answer 10

“Includes inequality” or “encompasses inequality” means that the answer will cover all or more of the range from the question stem. The trick answer will cover slightly less than the range in the stem.