StopLearnStopLearnIn mathematics we use the equals sign, =, to show that quantities are the same. However, very often, quantities are different, or unequal. For example, a mother is always older than her child their ages are always different. We say that there is inequality in their ages. This chapter explains the use of inequalities in arithmetic, algebra and in everyday life. It also introduces the inequality symbols.
Greater than, less than
The sum 5 + 3 = 8 is a simple equality. However, as we know, quantities are often not equal. For example:
5 + 5 ≠ 8
where ≠ means ‘is not equal to’. We can also write:
5 + 5 > 8
where > means ‘is greater than. Similarly we can write the following:
3 + 3 ≠ 8
3 + 3 < 8
where < means ‘is less than’.
≠, >, < are inequality symbols. They tell us that quantities are not equal. The > and < symbols are more helpful than ≠. They tell us more. For example, x ≠ 0 tell us that x does not have the value 0; x can be any positive or negative number. However, x<0 tell us us that x is less than 0; x must be a negative number.
Assessment
Answer the following questions
The symbols can be used to change word statements into algebraic statements. See examples below.
Let the distance between the villages be d km. Then, d > 18
A statement like d > 18 is called an inequality.
I spend N200 out of x naira.
Thus I have x – 200 naira left.
Thus x – 200 < 50.
Not greater than, not less than
In most towns there is a speed limit of 50 km/h. If a car, travelling at s km/h, is within the limit, then s is not greater than 50. If s < 50 or if s = 50, the speed limit will not be broken. This can be written as one inequality:
s ≤ 50
where ≤ means ‘less than or equal to’. Thus, not greater than means the same as less than or equal to.
In most countries, voters in elections must not be less than 18years of age. If a person of age a years is able to vote, then a is not less than 18. The person can vote if a > 18 or if a = 18. This can be written as one inequality:
a ≥ 18, where ≥ means ‘is greater than or equal to’. Thus, not less than means the same as greater than or equal to.
Graphs of inequalities
Linear inequalities
Inequalities like 3x > – 12 and 2x – y ≤ 7 have unknowns, or variables, with an index of 1 9i.e. x = x1 and y = y1 ). Inequalities with variables of index 1 are called linear inequalities.
3x > – 12 is a linear inequality in one variable (x); 2x – y ≤ is a linear inequality in two variables (x and y). This chapter is restricted to linear inequalities in one variable.
Linear Inequalities in one variable
When working with linear equations involving one variable whose highest degree (or order) is one, you are looking for the one value of the variable that will make the equation true. But if you consider an inequality such as x + 2 < 7, then values of x can be 0, 1, 2, 3, any negative number, or any fraction in between. In other words, there are many solutions for this inequality. Fortunately, solving an inequality involves the same strategies as solving a one variable equation. So even though there are an infinite number of answers to an inequality, you do not have to work any harder to find the answer. To review how to solve one variable equations,
However, there is one major difference that you must keep in mind when working with any inequality. If you multiply or divide by a negative number, you must change the direction of the inequality sign. You’ll see why this is the case soon.
Let’s go back and look at x + 2 < 7. If this were an equation, you would only need to subtract 2 from both sides to have x by itself.
x + 2 < 7
– 2 – 2
…………….
x < 5
Keep in mind that the new rule for inequalities only applies to multiplying or dividing by a negative number. You can still add or subtract without having to worry about the sign of the inequality.
But what would happen if you had -2x ≥ 10? Before solving, If you let x = -5 or -6 or any other value that is less than -5, then the inequality will be true. So you would write your solution as x ≤ – 5. In the process of solving this inequality using algebraic methods, you would have something that looks like the following:
-2/-2 10/-2
x -5
Let’s Practice
2x + 3 > -11
Begin by getting the variable on one side by itself by subtracting 3 from both sides. Then divide both sides by 2. Since you are dividing by a positive 2, there is no need to worry about changing the sign of the inequality.
2x + 3 > -11
2x > -14
x > -7
4 – 3x 20
The solution to this problem begins with subtracting 4 from both sides and then dividing by -3. As soon as you divide by -3, you must change the sign of the inequality.
4 – 3x 20
-3x 16
x -16/3
5x – 7 > 3x + 9
Thissolution will require a little more manipulation than the previous examples. You have to gather the terms with the variables on one side and the terms without the variables on the other side.
5x – 7 > 3x + 9
-3x -3x
2x – 7 > 9
+7 +7
2x > 16
x > 8
There is another type of inequality called a double inequality. This is when the variable appears in the middle of two inequality signs. This is simply a shortcut way of writing two separate inequalities into one and using a shorter process for finding the solution.
Basic Rules of Inequalities
Rule 1
If a > b then b < a, i.e. if a is greater than b then b is less than a, If a < b then b > a and if a is less than b then b is greater than a
Rule 2
If a > b and b > e then a > e, e. g. if 6 > 4 and 4 > 2 then 6 > 2, If a < band b < e then a < e, e. g. if 3 < 7 and 7 < 10 then 3 < 10
Rule 3
If a > b then a + e > b + c or a – c > b – c, If a < b then a + e < b + c or a – c < b – c
i.e. we can add to or subtract from both sides of an inequality the same quantity without changing the sense (or sign) of the inequality.
Rule 4
If a > b and c is a positive number, i.e. c > 0 then ac > bc and a/c > b/c, If a < b and c > 0 then ac < bc and a/c < b/c
i.e. both sides of an inequality can be multiplied or divided by the same positive number without changing the sense of the inequality.
Rule 5
If a> b and c is negative i.e. c < 0 then ac < bc and a/c <b/c, If a < b and c < 0 then ac > bc and a/c > b/c
Note: Both sides of an inequality can be multiplied or divided by a negative number, but the sense of the inequality is reversed.
The sense of an inequality is changed if both sides are multiplied or divided by the same negative number.
Rule 6
If a > b and c > d then adding the inequalities a + c > b + d, If a < b and c < d then a + c < b + d
i.e. Inequalities having the same sense can be added side by side to each other without changing the sense of the inequalities.
Rule 7
If a > b and c > d then either a – c > b – d or c – a > d – b is true but not the two of them are true at the time.
Similarly if a < b and c < d then either a – c < b – d or c –a < d – b is true but not the two of them.
Rule 8
If a > b > 0 and c > d > 0 or a < b < 0 and c < d < 0 then ac > bd
Rule 9
If a > b and n > 0 then an > bn
e.g. 5 > 3 and 2 > 0
52 > 32 i.e.25 > 9
If a > b and n < 0 then an < bn
If a< b and n > 0 then an < bn
If a < b and n < 0 then an > bn
e.g. 4 < 6 and -2 < 0
4-2 > 6-2
i.e. 1/16 > 1/36
Example
-2 6x – 1 10
The strategy for solving this inequality is not that much different than the other examples. Except in this case, you are trying to isolate the variable in the middle rather than on one side or the other. But the process for getting the x by itself in the middle, you should add 1 to all three parts of the inequality and then divide by 6.
-2 6x – 1 10
-1 6x 11
-1/6 x 11/6
Graphing One-Variable Inequalities
Before graphing linear inequalities, we summarized below the different forms of inequalities, with its corresponding interval form and graph:
Let’s take a look at the inequality symbols and their meanings again.
There are just a few important concepts that you must know in order to graph an inequality. Let’s review a number line.
The negative numbers are on the left of the zero and the positive numbers are on the right.
Example
r > −5
This is read as “r is greater than -5.” This means it includes all numbers greater than, or to the right, of -5 but does not include -5 itself. We will have to show this by using an open circle and having the arrow shoot out to the right.
Example 2
x ≤ 0.4
This is read as “x is less than or equal to 0.4.” This time we include the 0.4 by using a closed circle and the arrow will shoot out to the left. The number 0.4 is in between the 0 and the 1 on a number line.
Here is a summary of the important details in graphing inequalities.
Make sure you read the inequality starting with the variable!
“greater than” or “greater than or equal to” – arrow shoots out to the right
“less than” or “less than or equal to” – arrow shoots out to the left will have open circles
≤ and ≥ will have closed circles.
Questions
Let’s try a couple examples.
Solution of inequalities
Balance method
Consider a compound in which 23 people live. T any one time there may be x people in the compound. If all 23 people are in the compound, then x = 23. This is an equation.
If some people have left the compound, then x < 23. This is the inequality.
The equation has only one solution: x = 23. The inequality has many solutions: if x < 23, then x could be 0, 1, 2, 3, . . ., 20, 21, 22.
Notice that negative and fractional values of x are impossible in this example.
Inequalities are solved in much the same way as equations. We use the balance method.
However, there is one important difference to be shown later on.
Example
Solve the inequality x + 4 < 6.
x + 4 < 6
subtract 4 from both sides.
x + 4 – 4 < 6 – 4
x < 2
x < 2 is the solution.
When solving inequalities, we do not normally try to list the values of the unknown.
Exercise
Example
Find the values of x that satisfy the inequality 3x – 3 > 7, such that x is an integer.
Note: an integer is any whole number
-3, 0, 22 are examples of integers.
3x – 3 > 7
Add 3 to both sides.
3x > 10
Divide both sides by 3.
x > 31/3
But x must be an integer.
Thus x can have values 4, 5, 6, ….
x = 4, 5, 6, … is the solution.
Multiplication and division by negative numbers
Consider the following true statement: 5 > 3. Multiply both sides of the inequality by -2.
This gives -10 > -6.
But this is a false statement. In fact, -10 is less than -6.
Similarly, dividing both sides of 15 > -12 (true) by -3 gives – 5 > 4 (false, since -5 < 4).
In general, when multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign to keep the statement true.
For example if -2x ≥ 14 is true, then dividing both sides by -2 gives the equivalent true statement x ≤ -7.
Example
Solve 5 – x > 3
Either
5 – x > 3
Subtract 5 from both sides.
-x > -2
Multiply both sides by -1 and reverse the inequality.
(-1) X (-x) < (-1) X (-2)
x < 2
or:
5 – x > 3
Add x to both sides.
5 > 3 + x
Subtract 3 from both sides.
2 > x
thus, x < 2
The second method in Example above shows that the rule of reversing the inequality sign when multiplying by a negative number is correct.
A linear inequality involves the relationship between linear functions, just as do linear equations. The difference, however, is that linear inequalities relate two linear functions using the symbols <, ≤, >, or ≥, which correspond respectively to less than, less than or equal to, greater than, and greater than or equal to. Because these relationships do not involve a strict equality, solutions for expressions that contain them are more complex than similar expressions that do involve strict equality. For the most part, the same rules we have used for linear equations also apply to linear inequalities-a few nuances must be considered, however.
Let’s consider a simple linear inequality: f(x) > 2x – 1. This expression simply means that the output of the function f are those values that are greater than 2x – 1. Let’s look at this inequality graphically–first, we must plot the line 2x – 1, which is the boundary between values that satisfy the inequality and those that do not.
The line in the above graph defines the function (or equation) f(x) = 2x – 1; what we want to find, however, is f(x) > 2x – 1. This is to say, the values of f(x) that satisfy this inequality are those that are larger than those corresponding to the expression 2x – 1. Since f(x) and 2x – 1 are both sets of values along the vertical axis, the plot of f(x) > 2x – 1 must be all values above the line, but not including the line. (If the expression was f(x) ≥ 2x – 1, then the graph would include the line.) To display this set of values for f(x) on the graph, we shade the region above the line, but we make the line broken to indicate that the line is not part of the region that satisfies the expression f(x) > 2x – 1.
If the inequality was f(x) < 2x – 1 or f(x) ≤ 2x – 1, then the region below the line 2x – 1 would be shaded (and the line would or would not be solid depending on which expression was used).
Practice Problem: Graph the inequality k(z) ≤ –z + 4.
Solution: First, let’s draw an appropriate set of axes and graph the line –z+ 4. We’ll use a solid line because the line itself is part of the solution set (since ≤ is used). Next, we’ll shade the area below the line. The result should look like the following graph.
We can perform a “necessary but not sufficient” check of the result to give us more confidence in this answer: pick any point in the shaded region and use those values to see if the inequality holds for that point. Let’s try the point (4, 0):
–z + 4 = –4 + 4 = 0
k(4) = 0 ≤ 0 Inequality holds
If the linear inequality is expressed entirely in terms of the independent variable, then the solution is all values on one side or the other of some vertical line x = c, where c is some constant. For instance, let’s look at the inequality –7x – 5 ≤ –2 + x. We can manipulate inequalities using the same rules of algebra we apply to algebraic equations. There is one difference, however: if both sides of an inequality are multiplied by a negative number, you must reverse the direction of the inequality. We can see this with simple numbers: 4 < 6, yet –4 > –6. Let’s express this rule generally for functions f(x) and g(x) and a constant value –c (where c > 0):
f(x) < g(x) → –cf(x) > –cg(x)
f(x) > g(x) → –cf(x) < –cg(x)
f(x) ≤ g(x) → –cf(x) ≥ –cg(x)
Interested in learning more? Why not take an online class in Algebra?
f(x) ≥ g(x) → –cf(x) ≤ –cg(x)
Now, let’s return to our example.
–7x – 5 ≤ –2 + x
–7x – 5 – x ≤ –2 + x – x
–8x – 5 ≤ –2
–8x – 5 + 5 ≤ –2 + 5
–8x ≤ 3
Now, recall the new rule:
–8x ≤ 3
(-8x) ≥ (3)
x ≥
This is the solution to the inequality. Note, however, that this is a range of values, not just a single value. We call the set of solutions to an equation or inequality the solution set. Let’s take a graphical look at the solution set for this inequality. Note that because our solution involves only the independent variable, we can plot the results using a number line instead of a planar graph. When doing so, use an open circle () if the endpoint of the solution set is not included, and use a closed circle () if it is included. We use an arrow (a ray) to indicate the solution set.
Note that we use a solid line because the inequality is of the form “greater than or equal to.” The shaded region in the graph, including the solid line, is the solution set of the inequality –7x – 5 ≤ –2 + x.
Practice Problem: Find and graph the solution set of 3x – 4 < 1 – x.
Solution: First, we can manipulate the inequality to find a corresponding solution set in terms of the independent variable x.
3x – 4 < 1 – x
3x – 4 + 4 < 1 – x + 4
3x < 5 – x
3x + x < 5 – x + x
4x < 5
x < 5/4
We can check this result by using a value that satisfies x < 5/4; let’s try x= 0.
3(0) – 4 < 1 – (0)
–4 < 1 Inequality holds
Now, let’s graph the result. Note that we use an open circle at x = 5/4 because the solution set is a strict inequality (the < symbol is used).
Read our disclaimer.
AD: Take Free online baptism course: Preachi.com 