Ch 6 - Inequalities & Absolute Values

Must Know Concepts

When have to divide or multiply an inequality by a negative number, you…

you must reverse the sign

When do you reverse the inequality sign?

When you divide or multiply by a negative number

How do you add inequalities together?

You need to check the direction the signs are facing. If they’re facing the same direction, you can add them

When can you add inequalities together?

When the signs are facing the same direction

How do you solve problems with an equation and inequality?

Solve for a variable and plug the equation into an inequality

When dealing with a compound inequality you must…

make sure to do the operations to both sides of the inequality

You can only multiply or divide an inequality by a variable when…

you know the sign of the variable. you need to know if you need to reverse the inequality sign or not

What method can you use to compare the size of multiple inequalities?

Make a number number line

If x² + 10 > 110, then which of the following could be the value of x?

8, -10, -12 (can select multiple answers)

-12

If x² > b and b is positive then, the√x is…

1. x is greater than the square root of b

2. x less than the negative of the square root of b

x > √b and x < √b

If x² < b and b is positive then, the√x is…

1. x is less than the square root of b

2. x less than the negative of the square root of b

If 2 absolute values are equal, it must be true that….

the expressions in the absolute values bars are either equal or opposite

the absolute value of the sum of 2 numbers, a and b, will always be _________ the absolute value of the 2 individual numbers?

| a + b | ??? |a| + |b|

less than or equal to

| a + b | ≤ |a| + |b|

If a and b are 2 non-zero numbers, and | a + b |= |a| + |b|, then |a| + |b|…

must have the same sign

|a - b| ??? |a| - |b|

|a - b| ≥ |a| - |b|

the absolute value of the subtraction of 2 numbers, a and b, will always be greater than the absolute value of the 2 individual numbers

If b ≠ 0 and |a-b| = |a| - |b|, then a and b must…

a and b must….

1. have the same sign

2. |a| ≥ |b|

if |x| < b, we know that…

1. x < b and x > -b

2. b < x < -b (other way to format it)

x would be between the 2 values on a number line

if |x| > b, we know that…

1. x > b

2. x < -b

When are there no solutions to an absolute value problem?

When the absolute value of an expression is equal to a negative number

When do you need to check an absolute value equation for extraneous solutions?

When there’s a variable on both sides of the absolute value equation

if |x| = |y|, what do we know about x?

either x = y or x = -y

We shouldn’t multiply or divide by a variable in inequality problem unless…

we know the sign of the variable (so that we know if we need to flip the inequality)

How are |a - b| and |a| - |b| ordered?

|a - b| ≥ |a| - |b|

|a - b| will always be greater than

If |a - b| = |a| - |b|, this means…

the second quantity is 0

or

both qualities (a and b) have the same sign

How are |a + b| and |a| + |b| ordered?

|a + b| ≤ |a| + |b|

|a + b| is less than

If |2x + 4| = 12, what are 2 possible values of x?

If 2x + 4 = 2x + 4, then 2x + 4 = 12 → 2x = 8 → x = 4

If 2x + 4 = -(2x + 4), then -2x - 4 = 12 → -2x = 16 → x = -8

4 and -8

if x² = |x|, then x must be…

1, 0, or -1

if x is not one of the 3 values then x² > |x| or x² < |x|

if x² < |x|, then x must be…

x must be between 0 and -1

-1 < x < 0

bc if x² = |x|, x can only be 0,1, or -1

if x² > |x|, then x must be…

x < -1 or x > 1

Solve | x + 1 | = -1

you can’t. an absolute value is equal to a negative number which isn’t possible