INEQUALITIES AND ABSOLUTE VALUES

What does an open circle and closed circle represent on a number line?

Open circle represents greater than or less than, not equal to

Closed circle represents greater than or equal to, less than or equal to

When must the inequality sign be reversed?

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

Is it okay to multiply or divide an inequality by a variable if it’s value isn’t known?

No, don’t multiply or divide an inequality by a variable unless the sign of the variable unless the sign of the variable is known.

Sqrt (x²) = ?

|x|

If a <= x <=b and c <= y <= d, how do we evaluate the minimum and maximum values of xy?

We evaluate the following 4 quantities = ac, ad, bc, and bd. Largest value will be the maximum and the smallest value will be the minimum

Calculate the value of x for |16x + 14| = |8x + 6|

If two absolute values are equal, the expressions within the absolute value bars are either equal or opposite.

Equal

16x + 14 = 8x + 6

8x = -8

x = -1

Opposite

16x + 14 = -8x -6

24x = -20

x = -5/6

Is it true that the absolute value of the sum of 2 numbers, a and b, will always be less than or equal to the sum of the absolute values of the two numbers?

Yes!
| a + b | <= | a | + | b |

Is it true that the absolute value of the difference of 2 numbers, a and b, will always be greater than or equal to the absolute value of a minus the absolute value of b?

Yes!

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

If the absolute value of an expression is equal to a negative number, will there be any solutions to that equation?

No! If the absolute value of an expression is equal to a negative number, there will no solutions to that equation

What is the arithmetic property of integers?

• If r is the remainder when the positive integer x is divided by 4, then 0 ≤ r < 4, so the maximum value of r is 3.

• If R is the remainder when the positive integer x is divided by 9, then 0 ≤ R < 9, so the maximum value of R is 8. Thus, the maximum value of r ² + R is 3² + 8 = 17.