Inequalities Flashcards
tells us that the value of one quantity is not the same as the other quantity.
Inequality
There are five symbols that are usually used to indicate inequality:
theless thansign (<)
thegreater thansign (>)
thegreater than or equalsign (≥)
theless than or equalsign(≤)
theunequalsign (≠)
theless thansign (<)
thegreater thansign (>)
thegreater than or equalsign (≥)
theless than or equalsign(≤)
theunequalsign (≠)
These symbols are called
Inequality sign
Suppose the inequalityx + 1 > 10.
(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!).
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.
Trichotomy Property of Inequality
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.
Reversal Property of Inequality
Example:Apply the reversal property to the following:
9 > -1
x < y
a > b
Solution:
-1 < 9
y > x
b<a
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.
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 <,≥,≤)
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.
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 <,≥,≤)
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.
Multiplication Property of Inequality (MDI)
If a > b, then ac > bc when c > 0 and ac < bc when c < 0
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.
Division Property of Inequality (DPI)
Example:Which of the following is a linear inequality in one variable?
a) 2x + 3y > -1
b) 5x – 1 < -8
c) x²+ 3x > -1
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.
try to solvex – 4 > 2
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.
Example:Solve for the inequality x + 9 > 10
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.
Example 1:Solve for the inequality x + 5 > 17 using the transposition method.
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.
Example 2:What is the largest whole number that will satisfy x – 5 < 90?
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.
Example 3:What is the smallest whole number that will satisfy the inequality x – 12 > 100?
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.
Example 1:Let us try to solve 2x + 1 < 9 .
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.
Example 2:Solve for the inequality 5x – 8 < 12
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.
Example 3:Let us try to solve -3x + 4 > 22
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.
Example 4:Solve the inequality 16 – 8x≤80
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.
Example 1:Solve the inequality 15 < 3 + 2x
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.
Example 2:Solve the following inequality:
3x/2<8
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.
Example 3:What is the smallest whole number that will satisfy the inequality x + 4 < 2x – 7?
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.
