linear inequalities

What are inequalities?

The open sentence which involves >, ≥, <, ≤ sign are called an inequality. Inequalities can be posed as a question much like equations and solved by similar techniques step-by-step.

What are inequation?

A statement indicating that value of one quantity or algebraic expression which is not equal to another are called an inequation.

For example;

(i) x < 5
(ii) x > 4
(iii) 5x ≥ 7
(iv) 3x - 2 ≤ 4

Thus, each of the above statements is an inequation.

Linear Inequations:

An inequation which involves only one variable whose highest power one is known as a linear inequation in that variable.

Linear inequation looks exactly like a linear equation with inequality sign replacing the equality sign.

The statements of any of the forms ax + b > 0, ax + b ≥ 0, ax + b < 0, ax + b ≤ 0 are linear inequations in variable x, where a, b are real numbers and a ≠ 0.

For example;

(i) 2x + 1 > 0,
(ii) 5x ≤ 0,
(iii) 5 - 4x < 0,
(iv) 9x ≥ 0

Thus, each of the above statement is linear inequation in variable x.

Domain of the variable or the Replacement set:

For a given inequation, the set from which the values of the variable are replaced is called domain of the variable or the replacement set.

For example;

1. Consider an inequation x < 4. Let the replacement be the set of whole numbers (W).

Solution:

We know that W = {0, 1, 2, 3, ...}. We replace x by some values of W. Some values of x from W satisfy the inequation and some don’t. Here, the values 0, 1, 2, 3 satisfy the given inequation x < 4 while the other values don’t.

Thus, the set of all those values of variables which satisfy the given inequation is called the solution set of the given inequation.

Note:

Every solution set is a subset of replacement set.
Therefore, the solution set for the inequation x < 4 is S = {0, 1, 2, 3} or S = {x : x ∈ w, x < 4}

2. Consider an inequation x < 5. Let the replacement set be the set of natural numbers (N). Solution:

We know that N = {1, 2, 3, 4, 5, 6, ...}. We replace x by some values of N which satisfy the given inequation. These values are 1, 2, 3, 4.

Thus, a solution set of all those values of variables which satisfy the given inequation is called the solution set of the given inequation.


Note:

Every solution set is a subset of replacement set.
Therefore, the solution set for the inequation x < 5, x ∈ N is S = {1, 2, 3,} or S {x : x ∈ N, x < 5}.

3. Find the replacement set and the solution set for the inequation x ≥ -2 when replacement set is an integer.

Solution:

Replacement set = {... -3, -2, -1, 0, 1, 2, 3, ...}

Solution set = {-2, -1, 0, 1, 2, ...} or S = {x : x ∈ I, x ≥ -2}

4. Find the solution set for the following linear inequations.

  (i) x > -3 where replacement set is S = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

  (ii) x ≤ -2 where replacement set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4}

Solution:

(i) Solution set S = {-2, -1, 0, 1, 2, 3, 4} or S = (x : x ∈ I, -3 < x ≤ 4}

(ii) Solution set S = {-2, -3, -4, -5} or S = {x : x ∈ I,- 5 < x ≤ - 2

Solving linear inequalities is almost exactly like solving linear equations.

• Solve x + 3 < 0.

If they'd given me "x + 3 = 0", I'd have known how to solve: I would have subtracted 3 from both sides. I can do the same thing here:

Then the solution is:

x < –3

In all, we have seen four ways, with a couple variants, to denote the solution to the above inequality:

notation	format	pronunciation
inequality	x < –3	x is less than minus three
set	i) {x | x is a real number, x < –3} ...or:  ii) {x | x < –3}	i) the set of all x, such that x is a real number and xis less than minus three ii) all x such that x is less than minus three
interval		the interval from minus infinity to minus three
graph	either of the following graphs:

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Here is another example, along with the different answer formats:

• Solvex – 4 > 0.

If they'd given me "x – 4 = 0", then I would have solved by adding four to each side. I can do the same here:

Then the solution is: x > 4

Just as before, this solution can be presented in any of the four following ways:

notation	format	pronunciation
inequality	x > 4	x is greater than or equal to four
set	i) {x | x is a real number, x > 4} ...or:  ii) {x | x> 4}	i) the set of all x, such that x is a real number, and x is greater than or equal to four ii) all x such that x is greater than or equal to four
interval		the interval from four to infinity, inclusive of four
graph	either of the following graphs:

Regarding the graphs of the solution, the square bracket notation goes with the parenthesis notation, and the closed (filled in) dot notation goes with the open dot notation. While your present textbook may require that you know only one or two of the above formats for your answers, this topic of inequalities tends to arise in other contexts in other books for other courses. Since you may need later to be able to understand the other formats, make sure now that you know them all. However, for the rest of this lesson, I'll use only the "inequality" notation; I like it best. 

Solving Linear Inequalities:
     Elementary Examples

• Solve2x < 9.

If they had given me "2x = 9", I would have divided the 2 from each side. I can do the same thing here:

Then the solution is: x < ⁹/₂
...or, if you prefer decimals (and if your instructor will accept decimal equivalents instead of fractions):

x < 4.5

• Solve ^x/₄ > ¹/₂.

If they had given me " ^x/₄ = ¹/₂ ", I would have multiplied both sides by 4. I can do the same thing here:

Then the solution is: x > 2

• Solve –2x < 5.

Remember how I said that solving linear inequalities is "almost" exactly like solving linear equations? Well, this is the one place where it's different. To explain what I'm about to do, consider the following:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

3 > 2
What happens to the above inequality when I multiply through by –1? The temptation is to say that the answer will be "–3 > –2". But –3 is not greater than –2; it is in actuality smaller. That is, the correct inequality is actually the following:

–3 < –2
As you can see, multiplying by a negative ("–1", in this case) flipped the inequality sign from "greater than" to "less than". This is the new wrinkle for solving inequalities:

When solving inequalities, if you multiply or divide through by a negative, you must also flip the inequality sign.
To solve "–2x < 5", I need to divide through by a negative ("–2"), so I will need to flip the inequality:

Then the solution is: x > ^–5/₂

• Solve ^{(2x – 3)}/₄  < 2.

First, I'll multiply through by 4. Since the "4" is positive, I don't have to flip the inequality sign:
^{(2x – 3)}/₄   < 2
(4) × ^{(2x – 3)}/₄  < (4)(2)
2x – 3 < 8
2x < 11
x < ¹¹/₂  = 5.5

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• Solve 10 < 3x + 4 < 19.

This is what is called a "compound inequality". It works just like regular inequalities, except that it has three "sides". So, for instance, when I go to subtract the 4, I will have to subtract it from all three "sides".
10 < 3x + 4 < 19
6 < 3x < 15
2 < x < 5

Solving Linear Inequalities:
     Advanced Examples

• The velocity of an object fired directly upward is given by V = 80 – 32t, where t is in seconds.
  
  When will the velocity be between 32 and 64 feet per second?
I will set up the compound inequality, and then solve for t:

    32 < 80 – 32t < 64
    32 – 80 < 80 – 80 – 32t < 64 – 80
    –48 < –32t < –16
    ^–48 / _–32  >  ^–32t / _–32  >  ^–16 / _–32
    1.5 > t > 0.5

Note that, since I had to divide through by a negative, I had to flip the inequality signs. Note also that you might (as I do) find the above answer to be more easily understood if written the other way around:

0.5 < t < 1.5
Looking back at the original question, it did not ask for the value of the variable "t", but asked for the times when the velocity was between certain values. So the actual answer is:

The velocity will be between 32 and 64 feet per second between 0.5 seconds after launch and 1.5 seconds after launch.

Always remember when doing word problems, that, once you've found the value for the variable, you need to go back and re-read the problem to make sure that you're answering the actual question. The inequality "0.5 < t < 1.5" did not answer the actual question regarding time. I had to interpret the inequality and express the values in terms of the original question.

• Solve 5x + 7 < 3(x + 1). 

First I'll multiply through on the right-hand side, and then solve as usual:

• 5x + 7 < 3(x + 1)
  5x + 7 < 3x + 3
  2x + 7 < 3
  2x < –4
  x < –2
  Since I divided through by a positive "2" to get the final answer, I didn't have to flip the inequality sign.
  
  
• You want to invest $30,000. Part of this will be invested in a stable 5%-simple-interest rate account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with 7% interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income. What is the least you can "invest" with your father, and still (assuming he really pays you back) get at least $1900 in interest?

First, I have to set up equations for this. The interest formula for simple interest is I = Prt, where I is the interest, P is the beginning principal, r is the interest rate expressed as a decimal, and t is the time in years. Since no time-frame is specified for this problem, I'll assume that t = 1. I'll let "x" be the amount that I'm going to "invest" with my father. Then there will be 30000 – x left to invest in the safe account. The interest on the business investment, assuming that I get paid back, will be:

(x)(0.07)(1) = 0.07x
The interest on the safe investment will be:

(30 000 – x)(0.05)(1) = 1500 – 0.05x
Then the total interest is:

0.07x + (1500 – 0.05x) = 0.02x + 1500
I need to get at least $1900, so:

0.02x + 1500 > 1900
0.02x > 400
x > 20 000
That is, I will need to "invest" at least $20,000 with my father in order to get $1,900 in interest income. Since I want to give him as little money as possible, I will give him the minimum amount:

I will invest $20,000 at 7%.

• An alloy needs to contain between 46% copper and 50% copper. Find the least and greatest amounts of a 60% copper alloy that should be mixed with a 40% copper alloy in order to end up with thirty pounds of an alloy containing an allowable percentage of copper.

This is similar to a mixture word problem, except that this will involve inequality signs, rather than "equals" signs. I'll set it up the same way, though:

	pounds	% copper	pounds copper
60%	x	0.6	0.6x
40%	30 – x	0.4	0.4(30 – x) = 12 – 0.4x
mix	30	between 0.46 and 0.5	between 13.8 and 15

How did I get those values in the bottom right-hand box? I multiplied the total number of pounds in the mixture (30) by the minimum and maximum percentages (46% and 50%, respectively). That is, I multiplied across the bottom row, just as I did in the "60%" row and the "40%" row, to get the right-hand column's value. The total amount of copper in the mixture will be the sum of the copper from the two alloys put into the mixture, so I'll add the expressions for the amount of copper from the alloys, and place the total between the minimum and the maximum allowable amounts of copper:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

13.8 < 0.6x + (12 – 0.4x) < 15
13.8 < 0.2x + 12 < 15
1.8 < 0.2x < 3
9 < x < 15
I will need to use between 9 and 15 pounds of the 60% alloy.

• Solve3(x – 2) + 4 > 2(2x – 3).

First I'll multiply through and simplify; then I'll solve:

3(x – 2) + 4 > 2(2x – 3)
3x – 6 + 4 > 4x – 6
3x – 2 > 4x – 6
–2 > x – 6            (*)
4 > x
x < 4

Why did I move the "3x" over to the right-hand side (to get to the line marked with a star), instead of moving the "4x" to the left-hand side? Because by moving the smaller term, I was able to avoid having a negative coefficient on the variable, and therefore I was able to avoid having to remember to flip the inequality when I divided off that coefficient. I find it simpler to work this way; I make fewer errors. But it's just a matter of taste.
Why did I switch the inequality in the last line and put the variable on the left? Because I'm more comfortable with inequalities when the answers are formatted this way. Again, it's only a matter of taste. The form of the answer in the previous line, "4 > x", is perfectly acceptable. As long as you remember to flip the inequality sign when you multiply or divide through by a negative, you shouldn't have any trouble with solving linear inequalities.