Inequalities

Question 1

tells us that the value of one quantity is not the same as the other quantity.

Answer 1

Inequality

Question 2

There are five symbols that are usually used to indicate inequality:

Answer 2

theless thansign (<)
thegreater thansign (>)
thegreater than or equalsign (≥)
theless than or equalsign(≤)
theunequalsign (≠)

Question 3

theless thansign (<)
thegreater thansign (>)
thegreater than or equalsign (≥)
theless than or equalsign(≤)
theunequalsign (≠)

These symbols are called

Answer 3

Inequality sign

Question 4

Suppose the inequalityx + 1 > 10.

Answer 4

(12) + 1 > 10

13 > 10TRUE

Furthermore,x = 15is also a solution since:

x + 1 > 10

(15) + 1 > 10

16 > 10TRUE

Also,x = 100is also a solution since:

x + 1 > 10

(100) + 1 > 10

101 > 10TRUE

Actually, there are a lot of possible values ofxthat will satisfy the inequalityx + 1 > 10.However, when solving an inequality, we do not list all these possible values ofx(since it will take us forever to do so!).

Question 5

This property tells us that a quantity is either larger than the other, smaller than the other, or equal to the other. It is mathematically impossible for two of these conditions to happen at once.

Answer 5

Trichotomy Property of Inequality

Question 6

This states that if we interchange the quantities on the left-hand and right-hand sides of the inequality, the sign of the inequality reverses.

For instance, we know that 5 > 2. Then, if we interchange the positions of 5 and 2, we must reverse the inequality sign to keep the inequality true.

Answer 6

Reversal Property of Inequality

Question 7

Example:Apply the reversal property to the following:

9 > -1
x < y
a > b

Answer 7

Solution:

-1 < 9
y > x
b<a

Question 8

According to this property, if we add or subtract the same number to both sides of the inequality, the inequality will still hold or the inequality will still be true.

Answer 8

Addition and Subtraction Property of Inequality (API/SPI)

If a > b, then a + c > b + c. Also, if a > b, then a – c > b – c (also applies with <,≥,≤)

Question 9

For instance, we know that 3 < 5. Suppose that we add 12 to both sides of the inequality:
3 + 12 < 5 + 12

15 < 17

Notice that the resulting inequality is still true.

Now, suppose that we subtract 12 to both sides of 3 < 5:

3 – 12 < 5 – 12

-9 < -7

Note that the resulting inequality is still true.

Answer 9

Addition and Subtraction Property of Inequality (API/SPI)

If a > b, then a + c > b + c. Also, if a > b, then a – c > b – c (also applies with <,≥,≤)

Question 10

tells us that if you multiply both sides of an inequality by the same positive number, the inequality holds. However, if you multiply both sides of the inequality by a negative number, the inequality sign is reversed to make the inequality hold.

Answer 10

Multiplication Property of Inequality (MDI)

If a > b, then ac > bc when c > 0 and ac < bc when c < 0

Question 11

If we divide both sides of the inequality with the same positive number, the inequality holds. However, if we divide both sides of the inequality with the same negative number, the inequality sign is reversed to make the inequality hold.

Answer 11

Division Property of Inequality (DPI)

Question 12

Example:Which of the following is a linear inequality in one variable?

a) 2x + 3y > -1

b) 5x – 1 < -8

c) x²+ 3x > -1

Answer 12

Solution:The only linear inequality in one variable is the one in letterb.It is the only inequality with one variable involved (which isx) and the exponent of that variable is 1.

Question 13

try to solvex – 4 > 2

Answer 13

To solve for the inequalityx – 4 > 2, we need to isolatexfrom other quantities. This means thatxmust be the only quantity on the left-hand side of the inequality.

The addition property of inequality allows us to add 4 to both sides of the inequality. Note that if we add 4 to both sides of the inequality, the -4 on the left-hand side will be eliminated and onlyxwill be remain.

x – 4 > 2

x – 4 + 4 > 2 + 4Addition Property of Inequality

x > 6

That’s it! We have isolatedxfrom other quantities. The solution set of the inequality isx > 6. This means that any number greater than 6 will satisfy the inequality.

Question 14

Example:Solve for the inequality x + 9 > 10

Answer 14

Solution:Again, we have to get rid of 9 on the left-hand side so that onlyxwill remain. To achieve this, we can subtract 9 from both sides of the inequality:

x + 9 > 10

x + 9 – 9 > 10 – 9Subtraction Property of Inequality

x > 1

Hence, the solution to the inequality isx > 1.

Question 15

Example 1:Solve for the inequality x + 5 > 17 using the transposition method.

Answer 15

Solution:We can transpose 5 to the right-hand side so thatxwill be the only quantity that will remain on the left-hand side (isolatexfrom other quantities). Note that 5 changes to -5 when transposed to the right-hand side.

x + 5 > 17

x > -5 + 17

x > 12

Thus, the solution to the inequality isx > 12or all real numbers greater than 12.

Question 16

Example 2:What is the largest whole number that will satisfy x – 5 < 90?

Answer 16

Solution:Rather than a solution, this example is asking for the largest number that will satisfy the inequality. So, our final answer should be a number and not a set.

But, we need to solve for the solution set of the inequality first to determine the largest number that satisfies the inequality.

x – 5 < 90

x < 5 + 90 Transposition Method

x < 95

This tells us that the solution set of the inequality is the set of all numbers that is less than 95. However, to solve the problem, we need to determine the largest whole number in the setx< 95. So,what is the largest whole number less than 95?That number is 94.

Note that 95 is not the largest number in the setx < 95since 95 is not included in this set.

Therefore, the answer to this example is94.

Question 17

Example 3:What is the smallest whole number that will satisfy the inequality x – 12 > 100?

Answer 17

Solution:Let us solve for the solution set first of the inequality using the transposition method:

x – 12 > 100

x > 12 + 100 Transposition Method

x > 112

The solution set we have obtained isx > 112or the set of all real numbers greater than 112. Now,what do you think is the smallest whole number in the set x > 112?That number is 113.

Note that the answer is not 112 since 112 is excluded inx > 112.

Therefore, the answer is113.

Question 18

Example 1:Let us try to solve 2x + 1 < 9 .

Answer 18

Solution:Again, we want to isolatexfrom other quantities and make it the only quantity remaining on the left-hand side. To achieve that, we can transpose 1 to the right-hand side:

2x + 1<9

2x < -1 + 9 Transposition Method

2x< 8

Now, we still have 2x on the left-hand side. We want it to bexonly. To get rid of the numerical coefficient 2, we have to divide both sides of the inequality by 2.

Thus, the solution set of the inequality isx < 4.

Question 19

Example 2:Solve for the inequality 5x – 8 < 12

Answer 19

Solution:

5x – 8 < 12

5x < 8 + 12Transposition Method

5x < 20

5x⁄5 < 20⁄5 Dividing both sides by 5 (DPI)

x < 4

Take note that there’s an important thing you have to consider when dividing both sides of an inequality. As per the division property of inequality, if you divide both sides of an inequality by a negative number, the inequality sign will be reversed.

Question 20

Example 3:Let us try to solve -3x + 4 > 22

Answer 20

Solution:We start by transposing 4 to the right-hand side:

-3x > -4 + 22

-3x > 18

Now, to makexthe only quantity on the left-hand side, we have to divide both sides by -3. After we divide both sides of the inequality by -3, the inequality sign will be reversed in accordance with the division property of inequality:

Thus, the answer isx < -6. This implies that any number less than -6 will satisfy the given inequality.

Question 21

Example 4:Solve the inequality 16 – 8x≤80

Answer 21

Solution:We start by transposing 16 to the right-hand side of the inequality:

16 – 8x≤80

-8x≤-16 + 80

-8x≤64

Now, our goal is to makexthe only quantity on the left-hand side and get rid of -8. To do it, we have to divide both sides of the inequality by -8. However, note that we have to reverse the inequality sign after the division process since we divide by a negative number.

-8x≤64

(-8x)/-8≤64/-8 Dividing both sides by -8

x≥-8 Inequality sign is reversed

The solution set isx≥-8.

Question 22

Example 1:Solve the inequality 15 < 3 + 2x

Answer 22

Solution:Our goal is to makexthe only quantity on one side of the inequality. Since the variablexis already on the right-hand side, our next move is to transpose 3 to the left-hand side of the inequality:

15 < 3 + 2x

-3 + 15 < 2x

12 < 2x

Now, to make thexthe only quantity on the right-hand side, we can divide both sides of the inequality by 2:

12 < 2x

12/2 < 2x/2

6 < x

Note that by the reversal property of inequality, we can make6 < xintox > 6.

Thus, the answer to this example isx > 6.

Question 23

Example 2:Solve the following inequality:

3x/2<8

Answer 23

Solution:Notice that the left-hand side of the inequality is fractional; it has a denominator of 2. We can remove the denominator by multiplying both sides of the inequality by 2:

Now, we have3x < 16. We can divide both sides of the inequality by 3:

Thus, the solution set of this inequality isx < 16/3.

This means that any number less than 16/3 will satisfy the given inequality.

Question 24

Example 3:What is the smallest whole number that will satisfy the inequality x + 4 < 2x – 7?

Answer 24

Solution:Again, to solve this inequality, our goal is to isolatexfrom other quantities. We can start by transposing 2x to the left-hand side:

Now, we can also transpose 4 to the right-hand side so that the only terms that will remain on the left-hand side are those that havexvariable only:

Simplifying and solving for the inequality:

The solution set isx > 11. The smallest whole number in this set is 12.

Thus, our answer for this example is12.

Question 25

try to solve word problems involving linear inequalities. Here are the steps:

Answer 25

Determine what is being asked in the problem.

Use a variable to represent the unknown in the problem.

Create a linear inequality that describes the given problem.

Solve the linear inequality and the given problem.

Question 26

Example 1:The sum of a number and 32 is less than or equal to 8. Determine the possible values of the number.

Answer 26

Solution:

1. Determine what is being asked in the problem.

The problem is asking us to find all the possible values of an unknown number such that the sum of that number and 32 is less than or equal to 8.

This means that the answer to this problem is not a single number but a set of numbers.

2.Use a variable to represent the unknown in the problem.

Letxrepresent the unknown number in the given problem.

3. Create a linear inequality that describes the given problem.

The problem states that the sum of the number and 32 is less than or equal to 8. Hence, we can write the inequality as follows:

x + 32≤8

4. Solve the linear inequality and the given problem.

Let us solve the inequality we have derived above.

x + 32≤8

x≤-32 + 8 Transposition Method

x≤-24

Thus, the solution set of the inequality isx≤-24. This means that all numbers less than or equal to -24 will satisfy the inequalityx + 32≤8.

Question 27

Example 2:Jim has PHP 200 for his science project. He is expecting to receive some money from his mother that will help him fund his project. Jim believes that he needs at least PHP 1000 to finish the project. Given this situation, what is the smallest amount of money that Jim’s mother should give so Jim can finish the project?

Answer 27

Solution:

1. Determine what is being asked in the problem.

The problem is asking us to find the smallest amount of money that Jim’s mother should give so that Jim can finish the project. Since we are looking for the smallest amount of money from the set of all possible amount of money that can satisfy the given condition, our answer to the problem should be a single number only.

2.Use a variable to represent the unknown in the problem.

Letxrepresent the amount of money that Jim’s mother will give.

3. Create a linear inequality that describes the given problem.

The problem states that Jim already has PHP 200 and he is expecting to receive additional money from his mother (which is represented by x). Thus, the total amount of money that Jim will have is x + 200.

To complete the project, Jim needs at least (greater than or equal to) PHP 1000. Thus, our inequality will be:

x + 200 ≥ 1000

4. Solve the linear inequality and the given problem.

Let us solve the inequality we have derived above.

x + 200 ≥ 1000

x≥-200 + 1000 Transposition Method

x≥800

Thus, the solution set of the inequality isx≥800.This means Jim’s mother should give any amount greater than or equal to 800 so Jim can finish the project.

However, we are looking for the smallest amount of money that Jim must receive from his mother and not the solution set. The smallest number inx≥800is 800 (800 is included in the solution set because of the equal sign).

Thus, the smallest amount of money that Jim’s mother should give isPHP 800.

Question 28

1) What is the smallest whole number that satisfies
2x – 9 > 11

a) 10
b) 11
c) 12
d) 13

Answer 28

B

Question 29

2) Solve for the inequality 10 – 3x < 2

a) x > 8/3
b) x < -8/3
c) x > 3/8
d) x < -3/8

Answer 29

A

Question 30

3) The sum of twice a number and 14 is at most 40. What is the largest possible value of the
number?

a) 12
b) 13
c) 14
d) 15

Answer 30

B

Question 31

4) Solve for the possible values of x in
𝑥/4 − 2 < 0

a) x > 8
b) x < 8
c) x > -8
d) x < -8

Answer 31

B

Question 32

5) The total number of books that Carl and Adam own is less than 85. Carl has 5 more books
than Adam. What is the largest possible number of books that Adam owns?

a) 35
b) 38
c) 39
d) 41

Answer 32

C