LINEAR INEQUALITIES - PROPERTIES, SOLVED EXAMPLES, EXERCISES

Linear inequalities in Basic Algebra,
if you have not already done so.

There we defined inequality as a true statement
containing the signs 'greater than' (>) or 'less than' (<).

We also saw that the vertex of inequality symbol
always lies towards small quantity.

We also saw how to write two or more inequalities
in one sentence, with example.

We also saw various other signs,
the inequality may contain with examples.

We also defined inequation as an open sentence
containing one of those signs.

We also understood set of roots or
solution set of an inequation.

We defined domain as the replacement set
of the variable of an inequation.

we also solved an example by trial and error.

Here we will solve more inequations
by applying the properties of inequations.

We may call Linear Inequations as Linear Inequalities.

Properties of Linear Inequalities :

Property 1 of Linear Inequalities :

Adding (or Subtracting) the same number to both sides of an Algebra Inequality does not change the order of the inequality sign ( i.e., '>', or '<').
i.e., if a < b then a + c < b + c and a - c < b - c
Similarly, if a > b then a + c > b + c and a - c > b - c
for any three numbers a, b, c.

Property 2 of Linear Inequalities :

Multiplying (or Dividing) both sides of an Algebra Inequality by the same positive number does not change the order of the inequality sign ( i.e., '>', or '<').
For any three numbers a, b, c where c > 0,
(i) if a < b then ac < bc and ac < bc
(ii) if a > b then ac > bc and ac > bc

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

The secrets will help you absorb, digest
and remember large chunks of information
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the Speed Study System.

Property 3 of Linear Inequalities :

Multiplying (or Dividing) both sides of an Algebra Inequality by the same negative number reverses the order of the inequality sign ( i.e., '>' to '<' and '<' to '>').
For any three numbers a, b, c where c < 0,
(i) if a < b then ac > bc and ac > bc
(ii) if a > b then ac < bc and ac < bc

Property 4 of Linear Inequalities :

If three numbers are related in such a way that the first is less (greater) than the second and the second is less (greater) than the third, then the first is less (greater) than the third.
This is called transitive property.

Property 5 of Linear Inequalities :

If a and b are of the same sign and a < b (a > b), then 1⁄a > 1⁄b (1⁄a < 1⁄b).
If reciprocals are taken to quantities of the same sign on both sides of an inequality, then the order of the inequality is changed.

Research-based personalized Math Help tutoring program : Linear Inequalities

Here is a resource for Solid Foundation in
Math Fundamentals from Middle thru High School.
You can check your self by the

FREE TRIAL.

Are you spending lot of money for math tutors to your
child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and
more personalized Math Help to identify and fix
the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend
and that too With a minuscule fraction of the amount
you spent on tutors with unconditional 100% money
back Guarantee, if you are not satisfied.

SUBSCRIBE, TEST, IF NOT SATISFIED, RETURN FOR FULL REFUND

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program
that identifies exactly where your child needs help and
then creates a personal instruction plan tailored to your
child’s specific needs.

If your child can use a computer and access
the Internet, he or she can use the program.
And your child can access the program anytime
from any computer with Internet access.

Unique program to help improve math skills quickly and painlessly.

There is an exclusive, Parent Information Page provides YOU
with detailed reports of your child’s progress so you can
monitor your child’s success and give them encouragement.
These Reports include

• Time spent using the program
• Assessment results
• Personalized remediation curriculum designed for your child
• Details the areas of weakness where your child needs additional help
• Provides the REASONS WHY your child missed a concept
• List of modules accessed and amount of time spent in each module
• Quiz results
• Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section
of the program, available exclusively to you, the parent.
The section is accessed by a password that YOU create and use.
No unauthorized users can access this information.

Personalized remediation curriculum designed for your child

Thus The features of this excellent Tutoring program are

• Using detailed testing techniques
• Identifing exactly where a student needs help.
• Its unique, smart system pinpointing precise problem areas -
• slowly and methodically guiding the student
• raising to the necessary levels to fix the problem.

Not a “one-size-fits-all” approach!

Its research-based results have proven that
it really works for all students! in improving
math skills and a TWO LETTER GRADE INCREASE in
math test scores!,if they invest time in using
the program.

Proven for More than 10,000 U.S. public school
students who increased their math scores.

Solving Linear Inequations using the Properties :

To solve linear inequation, collect terms containing unknown quantity (variable) on left side and constants on the right side.
Then reduce the coefficient of the unknown quantity to unity.
While doing this, remember the properties of the algebra inequalities.

Great Deals on School & Homeschool Curriculum Books

Set of Solved Examples: Linear Inequalities

Solved Example 1 : Linear Inequalities

Check whether x = 6 is a solution of 9x + 1 > 7x + 5

Solution:
When x = 6, the L.H.S. of the inequation = 9x + 1 = 9(6) + 1 = 55
and the R.H.S. of the inequation = 7x + 5 = 7(6) + 1 = 43
We know 55 > 43. So x = 6 satisfies the given inequation.
x = 6 is a solution of 9x + 1 > 7x + 5.

Solved Example 2 : Linear Inequalities

Check whether t = -3 is a solution of (7t + 5)⁄6 ≥ (t - 1)⁄2

Solution:
When t = -3,
the L.H.S. of the inequation = (7t + 5)⁄6 = {7(-3) + 5}⁄6 = -8⁄3
and the R.H.S. of the inequation = (t - 1)⁄2 = (-3 - 1)⁄2 = -2
We know -8⁄3 < -2. So t = -3 does not satisfy the given inequation.
t = -3 is not a solution of (7t + 5)⁄6 ≥ (t - 1)⁄2

Solved Example 3 : Linear Inequalities

If the domain of the variable is N (the natural number set),
solve the inequation (x + 2) ≤ (3x - 4)

Solution:
(x + 2) ≤ (3x - 4) ⇒ x - 3x ≤ -4 -2 ⇒ -2x ≤ -6
Dividing throughout by -3
(negative number division causes reversing of inequality sign), we get
x ≥ 3. Ans.
Since the domain is the set of Natural numbers,
x = { 3, 4, 5, 6, ..............}. Ans.

Solved Example 4 : Linear Inequalities

If the domain of the variable is Q (the set of rational numbers),
solve the inequation (2x - 3)⁄4 > (3x + 4)⁄3

Solution:
(2x - 3)⁄4 > (3x + 4)⁄3
Multiplying both sides by 12, we get
3(2x - 3) > 4(3x + 4) ⇒ 6x - 9 > 12x + 16 ⇒ 6x - 12x > 16 + 9
⇒ - 6x > 25
Dividing throughout by -6
(negative number division causes reversing of inequality sign), we get
x < -25⁄6. Ans.

Solved Example 5 : Linear Inequalities

If the domain of the variable is Q (the set of rational numbers),
solve the inequation 3t⁄4 - 2⁄3 > t⁄2 + 1⁄3

Solution:
3t⁄4 - 2⁄3 > t⁄2 + 1⁄3
Multiplying both sides by 12, we get
9t - 8 > 6t + 4 ⇒ 9t - 6t > 4 + 8 ⇒ 3t >12 ⇒ t > 4 . Ans.

Exercise : Linear Inequalities

1. Check whether y = 2 is a solution of (2y + 3)⁄3 < (3y - 4)
2. Check whether z = -5 is a solution of (z⁄5 - z⁄2) < 3
3. Solve the inequation -7x + 78 ≥ 3x - 72 where x is a variable on Z (the set of integers).
4. Solve the inequation 13y⁄6 - 5⁄2 ≥ y⁄3 + 73⁄6 where y is a variable on N (the set of natural numbers).
5. Solve the inequation 2x + 3 < 10, where x is a variable in Q (the set of rational numbers).

For Answers see at the bottom of the page.

Answers to Exercise : Linear Inequalities

(1) No (2) Yes. (3) x ≤ 15 (4) y ≥ 8 (5) x < 7⁄2

Progressive Learning of Math : Linear inequalities

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.