Lesson Solving simple and simplest linear inequalities

This Lesson (Solving simple and simplest linear inequalities) was created by by ikleyn(52754)

: View Source, Show
About ikleyn:

Solving simple and simplest linear inequalities

Problem 1

Solve an inequality x-6 > 2.

Solution

To solve inequality

x - 6 > 2,


add 6 to both sides.  You will get an equivalent inequality 


x - 6 + 6 > 2 + 6,    which is the same as  


x > 8.


It is your ANSWER: x > 8.  The solution is completed.

Problem 2

Solve an inequality 2x + 4 > -14.

Solution

Subtract 4 from both sides.  You will get an equivalent inequality

    2x > -14-4,  or,  which is the same,

    2x > - 18.


Now divide both sides by 2.  You will get an equivalent inequality

    x > -9.         


It is your ANSWER:  x > -9.  The solution is completed.

Problem 3

Solve an inequality -4x - 6 > 18.

Solution

-4x - 6 > 18  ====>  add 6 to both sides. You will get an EQUIVALENT inequality


-4x > 18 + 6,   which is the same as


-4x > 24.


Now divide both sides by the number (-4).  Since you divide both sides of the inequality by negative number,

you MUST change the inequality sign by the opposite one.  So you get


x < ,   which is the same as


x < -6.


It is your ANSWER:  x < -6.  The solution is completed.

Thus if you are given an inequality of the form

             ax + b < c         (1)

where "a", "b" and "c" are the numbers, you subtract the value of "b" from both sides, reducing the given inequality to the form

             ax < c.

Your next step is to divide both sides by the coefficient "a".

You must be careful at this step and remember that if a > 0, then you get an equivalent inequality

             x < c%2Fa

,
which is your answer.

But if a < 0, then with the division you must change the sign of the resulting inequality to the opposite one:

x > c%2Fa

.

That is all in this simple case.

If you have slightly more complicated inequalities of the form

ax + b < cx + d,

then your first step is to transform them to the standard form (1) and then to complete the solution as it was explained above.

See the following examples.

Problem 4

Solve an inequality 4x-6 < 2x + 3.

Solution

Add 6 to both sides.  You will get an equivalent inequality


4x < 2x + 9.


Subtract 2x from both sides.  You will get an equivalent inequality


2x < 9.


Divide both sides by 2.  You will get an equivalent inequality


x <  = 4.5,


which is your solution.


Answer.  x < 4.5,  or, in the interval form, the solution is  (,).

Problem 5

Solve an inequality 2*(3x-2) < 4*(x + 1).

Solution

Open parentheses


6x - 4 < 4x + 4.


Add 4 to both sides.  You will get an equivalent inequality


6x < 4x + 8.


Subtract 4x from both sides.  You will get an equivalent inequality


2x < 8.


Divide both sides by 2.  You will get an equivalent inequality


x < 4,


which is your solution.


Answer.  x < 4,  or, in the interval form, the solution is  (,).

Problem 6

Solve an inequality 4*(2-3x) < 3*(x + 1).

Solution

Open parentheses


8 - 12x < 3x + 3.


Subtract 8 from both sides.  You will get an equivalent inequality


-12x < 3x - 5.


Subtract 3x from both sides.  You will get an equivalent inequality


-15x < -5.


Divide both sides by (-15).  Since you divide by an negative number, change the inequality sign by the opposite one.
You will get an equivalent inequality


x >  = .


which is your solution.


Answer.  x > ,  or, in the interval form, the solution is  (,).

Problem 7

Solve an inequality and graph the solution set.
3x - 1 > x+5.

Solution

Your starting inequality is

    3x - 1 > x + 5


Add 1 to both sides.  You will get

    3x     > x + 6.


Subtract x from both sides.  You will get

    2x     >     6.


Divide both sides by 2

     x > 3.


It is the ANSWER :  the solution set is  { x > 3 },  or  (3,oo).


                     The plot is shown below



    -----------------|-----|-----|-----(=====|=====|=====|===============
    -oo              0     1     2     3     4     5     6             oo



          In the plot, the solution set is presented by bold line.

My other lessons on solving inequalities are
    - Solving absolute value inequalities
    - Advanced problems on solving absolute value inequalities
    - Solving systems of linear inequalities in one unknown
    - Solving compound inequalities

    - What number is greater? Comparing magnitude of irrational numbers
    - Arithmetic mean and geometric mean inequality
    - Arithmetic mean and geometric mean inequality - Geometric interpretations
    - Harmonic mean
    - Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b), and sqrt(c)

    - Solving problems on quadratic inequalities
    - Solving inequalities for high degree polynomials factored into a product of linear binomials
    - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials
    - Solving inequalities for rational functions with non-zero right side
    - Another way solving inequalities for rational functions with non-zero right side

    - Advanced problems on inequalities
    - Challenging problems on inequalities
    - Solving systems of inequalities in two unknowns graphically in a coordinate plane
    - Solving word problems on inequalities
    - Proving inequalities
    - Math circle level problem on inequalities
    - Math Olympiad level problems on inequalities
    - Entertainment problems on inequalities
under the topic Inequalities, trichotomy of the section Algebra-I.

My lessons on domains of functions are
    - Domain of a function which is a quadratic polynomial under the square root operator
    - Domain of a function which is a high degree polynomial under the square root operator
    - Domain of a function which is the square root of a rational function.
under the topic Functions, Domain of the section Algebra-I.

See also OVERVIEW of lessons on inequalities and domains of functions.

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

This lesson has been accessed 2848 times.