In this explainer, we will learn how to solve one-step linear inequalities by addition or subtraction.
An inequality is a mathematical sentence containing an inequality symbol
; it shows that the value of one
expression is greater than the value of another, while an equation states the equality of two
expressions. When a variable is involved in an inequality, solving an inequality means finding the
range of values that the variable can take to make the inequality true. This range of values is given
as an inequality in the form
where is a number.
Often, we come across more complex inequalities, for instance, . In this explainer, we are going to learn how to simplify this type of inequalities to
get an inequality of the form
,
where is a number.
Let us recall the balance method for equations. An equation states the equality between
two expressions; we can represent it with a balance scale. For instance, the equation
represented on the diagram is .
Now, we do not want to solve the equation but the inequality
. Not only do we know that is not equal to 8 but also that
is greater than 8. We can represent this on an unbalanced scale by considering that
weighs more than 8, so that its side is lower than the side where there is a lighter weight
(here 8).
Now, as with a real balance scale, if the same weight is removed from or added to each side
of the scale, the side on the left will
still be heavier than the side on the right.
We can write this formally as
that is,
We can summarize this by writing the addition and subtraction rule for inequalities.
Addition and Subtraction Rule for Inequalities
An inequality still holds true when the same number is added to or subtracted from both of its sides.
Let us look at some examples to see how to apply this rule to simplify inequalities.
Example 1: Finding an Equivalent Inequality by Using the Addition and Subtraction Rule
Complete using , , , or
: If ,
then .
Answer
We start here with and we want to find something about
. We know that we can add the same number to both sides of an
inequality and it will still hold true. So, we have
that is,
Hence, the inequality should be completed using .
Example 2: Solving an Inequality by Adding
Given that , what number do you need to add
both sides of the inequality to colve for
Answer
To solve , we need to have on its own on
one side. For this, we add 43 to both sides, so that we will have
on the left-hand side, while we have
on the right-hand side. This will give .
Hence, the answer is that we need to add 43 to both sides of the inequality.
Example 3: Solving an Inequality by Subtracting
If , then .
3
Answer
We start here with the inequality . To solve it for
, we need to have on its own on one side. For
this, we take away 6 from each side:
that is,
Example 4: Solving an Inequality by Subtracting
If , then .
Answer
We start here with the inequality . To solve for
, we need to have on its own on one side. For this, we take away 45
from each side:
that is,
Example 5: Finding an Equivalent Inequality Using the Addition and Subtraction Rule
Select the option that is equivalent to .
Answer
We start here with the inequality . To solve it for
, we need to have on its own on one side. For this, we take away 7
from each side:
that is,
We have solved our inequality; however, none of the options given in the questions
displays what we have found. We need to inspect the options more closely to realize that
one of them is equivalent to our solution; namely, (option D). Indeed, if we draw and on a number line so that
is less than , then we see that is greater than .
So, and are equivalent.
Hence, our answer is (option D).
Key Points
An inequality is a mathematical sentence containing an inequality symbol
; it shows that the value of one
expression is greater than that of another.
To rearrange an inequality of the form
to the form , we use
the addition and subtraction rule for inequalities.
The addition and subtraction rule for inequalities states that an inequality still holds
true when the same number is added to or subtracted from both of its sides.
If the number is less than the number , then
is greater than .
This means that the inequality is equivalent to
. With more complex inequalities, we have, for instance, that
is equivalent to .
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