Download Inequalities

Inequalities
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Inequalities
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Definition of an inequality
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Types of inequalities
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Basic notations
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Manipulation of basic operators on inequalities
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Conditional and identity equations
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Practice questions
Definition
Inequality is a statement about the relative size or order of the objects or expressions ( talking
mathematically) or if they are equal or not.
Types of inequalities
There are basically five types of inequalities or ordering functions especially when working with
real numbers. They are with their symbols like “<” (less than), “>” (greater than), “≤ (less than
or equal to), “≥ (greater than or equal to), “ ≠ “ ( not equal to ).
< and > symbols are considered to be strict inequality signs,
≤ and ≥ symbols are considered to be non – strict inequality signs.
The set that contains all the solutions of a particular inequality is called the solution set of the
inequality.
Equivalent inequalities are inequalities that have the same solution set.
To solve inequalities we use an approach that parallels the approach of solving equations. We
shall get an equivalent inequalities until we arrive at one whose solution set is clear.
Compound inequalities
A compound inequality is two or more inequalities joined together. To be a solution of an “or”
inequality, a value has to make only one part of the inequality true. To be a solution of an “and”
inequality, a value must satisfy both the parts of the inequalities true. For example,
X < 3 or x < 2
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The “and” inequality is often written as a double inequality like this,
1< x < 2
To solve an “or” type compound inequality, just solve the two sentences separately. To solve an
“and” type compound inequality, make sure that if you add a number or multiply a number,
you do it to all three parts of the inequality.
Notations
a < b means a is less than b
a > b means a is greater than b,
a ≤ b means a is less than or equal to b,
a ≥ b means a is greater than or equal to b,
a ≠ b means a is not equal to b, but doesn’t mention if a is less than or greater than b.
Additional notations
a << b means a is much less than b,
a >> b means a is much greater than b.
Manipulation of basic operators on inequalities
The rule for addition is same as inequalities as it is for an equation.
Transitivity rule
Considering a, b and c as real numbers,
If a > b an b > c, then a > c
If a < b and b < c then a < c
Let us consider three mathematical expressions say, a, b and c.
If a < b, then a + c < b + c,
If a > b, then a + c > b + c
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The inequality does not change when an equal quantity is added to both the sides.
The rule very well holds good for subtraction as well, because subtraction is an addition with
opposite or negative quantity.
If a < b, a - c < b – c
If a > b, a – c > b - c
But, multiplication rule is different,
Let us consider three mathematical expressions say, a, b and c.
If a<b and c is positive, then ac < bc
If a < b and c is negative, then ac > bc
This rule holds good for division as well, because division is considered as multiplying with an
inverse of a quantity.
If a < b, 1/a > 1/b
If a > b, 1/a < 1/b
Additive inverse
If a < b, then -a > - b
If a > b, then –a < -b
Triangle Inequality
For any triangle, the length of given side is less than or equal to other two sides but
greater than or equal to the difference between the two sides.
We studied the inequalities with one variable, below is an example of inequality with two
variables.
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ax + by < c
or ax - by > c
Absolute value inequality
Absolute value inequality is an inequality is an inequality which has an absolute value sign with
a variable inside. For example, |x – 3 | > 3.
Systems of linear inequalities
One linear inequality in two variables divides the plane into two half planes, consists of two
linear inequalities that that have the same variables. For example,
2x+3y < 4 and 3x+5y > 6, if x and y represent the same item in both the inequalities. The
solution consists of a region in the coordinate plane that satisfies both the inequalities.
Systems of inequalities are useful in finding combinations of products that would yield good
profits.
Conditional equations
Let us consider an equation x + 3 = 5, we say this equation is neither true nor false, because the
letter x can take any value.
Solving the equation, x =2 is a solution, hence 2 satisfies this equation.
Another value for x say, 8 if substituted in the equation does not satisfy the same, hence the
equation becomes false.
Such equation whose truth or falsehood depends on the value of the variable is called
conditional equations.
Identity equation
An equation that does not depend on the value of the variable is called an identity, for example,
5 ( x +2) – 5 = 5x + 5
This equation is always true.
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Practice questions
1. What is the range of values of x if
X 2≥ 169
2. John and his younger brother want to buy a gift for their mom. John agrees to pay $ 6 more
than his brother. If the gift does not cost more than $ 30, what is the greatest possible amount
paid by John?
A. X ≤ 17
B. X ≥ 25
C. X ≤ 18
D. X = 22
3. If 8 – 4y > 20, what is the value of y?
A. Y > -3
B. Y > 3
C. Y < -3
D. Y < 3
4. If 0 < x < 1, then which of the following is false?
A. 2/x < 2
B. X2 < x
C. 2x < 2
D. 3x < 3
5. An aptitude test has 20 multiple choice questions. A correct answer is given 3 marks and
every wrong answer leads to deduction of 1 mark. No marks are awarded or deducted for
questions not attempted. A boy attempted total of 18 questions and scored a total above 32
marks. Find the minimum number of correct answers he got.
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A. X < 12.5
B. X > 12.5
C. X > 15.4
D. X > 13
6. If a and y are real numbers, square of y I equal to square root of x, which of the following
must be true?
1. X = y4
2. X ≥ 0
3. Y ≥ 0
A. 1 only
B. 1 and 2 only
C. All three
D. 2 only.
7. Find the range of values of x for which the following inequalities are satisfied simultaneously.
6x – 3 ≤ 2x +17 and {(2x – 1)/3} – {(1 – 2x)/2} > -3.
A. -13/10 < X ≤ 5
B. -12/14 < x < 5
C. 12/5 ≤ x < 5
D. 13/10 < x ≤ 6
8. At a snack bar, a customer orders a soda gets a cup containing c ounces of soda, where c is at
least 12 but, no more than 13, which of the following describes all possible values of c?
A. 12 > c >13
B. 12 < c <13
C. 12 ≤ c < 13
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D. 12 ≥ c > 13
9. Determine the type of equation
4x – 3( x – 2 ) = x + 6
A. Identity equation
B. Conditional equation
C. Both
D. None of the above
10. Depict the regions which satisfy the inequalities given:
X + y < 3 and x – y > 2
11. A retiree requires an annual income of at least $1400 from an investment that earns
interest at 7.5% per year. What is the approximate smallest amount the retiree must invest in
order to achieve the desired amount.
A. $17000
B. $18666
C. $19432
D. $ 20000
Answers
1. 13 to -13
2. C
3. A
4. A
5. B
6. B
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7. A
8. B
9. B
10.
11. B
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