Inequalities

Question 1

Open circle in inequalities

Answer 1

will not include that variable but includes other than that

Question 2

Close circle in inequalities

Answer 2

will include that variable and other variables including that

Question 3

When you divide or multiply by a negative number

Answer 3

the inequality must change the sign

Question 4

In order to add/subtract and equalize something in the equalities, they have to be

Answer 4

facing the same sign
Example:
-x+7y > 26
-7y+2x > -12

Question 5

When we are multiplying or dividing by a variable

Answer 5

if the sign of the variable is unknown, we cannot know whether to rotate the inequality sign

-> so we should never multiply or divide an inequality by a variable unless the signs of the variable is known

Question 6

Compound inequality: a

Answer 6

You have to do it to all the sides

a a-5 < x+5-5 a-5 < x< b-5

Question 7

When comparing the relative size of the variables expressed in multiple inequalities

Answer 7

consider setting up a number line to compare the values graphically

Question 8

What is the possible value of

x^2>b

Answer 8

x < -sqrt(b)

x > sqrt(b)

Question 9

If a <= x <=b and

c <= y <= d , find the maximum value of xy

Answer 9

we evaluate the following four quantities ac, ad, bc and bd. The largest of the four values will be the maximum and the smallest one will be the minimum of xy

Question 10

For any real numbers a, if a => 0, |a| =

Answer 10

a

Question 11

For any real numbers a, if a <= 0, |a| =

Answer 11

-a

Question 12

We know that the absolute value mask the true value inside the bars could be positive or negative

Answer 12

thus the value inside the bars can be positive or negative

-> so we need to solve the equation twice -> first one where the value is post

Question 13

Equation that contain absolute value

Answer 13

we must isolate those absolute value bars and perform the work outside them prior to splitting the equation
i.e : 5 |2x-1| -5 = 20
      = 5 |2x-1|= 25
       =  |2x-1|= 5
         = 2x-1 = -5  
       and 2x-1 = 5

Question 14

|5| = |5| and

Answer 14

|-5| = |-5|

Question 15

|a+b|

Answer 15

<= |a|+|b|

Question 16

If a and b are non-zero number and |a+b| = |a| + |b| then a and b must have

Answer 16

same sign

-> either both negative or both positive

Question 17

|a-b| >=

Answer 17

|a|-|b|

Question 18

If p and q are non-zero number and |p-q| = |p| - |q| then

Answer 18

1) pq > 0 (says that not have to be the same sign)
2) p’s distance from zero on the # line is greater than q’s distance from zero on the # line –> so the number itself should be bigger
or. ..
1) the second quantity is 0

Question 19

Can you see the similarity between x^2 > 4 and |x| >4

Answer 19

yes two answers

Question 20

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

Answer 20

there is no solutions to that equation

|x+1| = -2 -> can’t have this

Question 21

If an absolute value has a variable on both sides of the equation, we must

Answer 21

check the solutions of the equations for the extraneous solutions
-> extraneous variable is one that appears to be the solution but is not

Question 22

If |6x+14| = |8x+6| then what is the product of all possible values of x

Answer 22

You can get all the possible values by solving these two
A. Equal: |6x+14| = |8x+6|
B. Opposite: |6x+14| = -|8x+6|

Question 23

0 is a multiple of

Answer 23

all numbers

Question 24

ab > ac

Answer 24

two possibilities for ab and ac, depending on what the sign of a is

1) b> c
2) b