Inequalities with Absolute Value

🏆Practice inequalities with absolute values

Inequality

When you come across signs like $<$ or ,$>$ you will know it is an inequality.
The result of the inequality will be a certain range of values that you will have to find.

An important rule to keep in mind: when you double or divide both sides of the operation, the sign of the inequality is reversed!

Absolute Value Inequality

We can solve absolute value inequalities in $2$ ways:
In the geometric method and in the algebraic method.

Test yourself on inequalities with absolute values!

Given:

$$\left|x-4\right|<8$$

Which of the following statements is necessarily true?

The geometric method

• We will draw a number line, and mark the point from which we are interested in the distance to $X$.
• Then we find the points whose distance from the relevant point is exactly the distance mentioned in the condition
• Now, let's pay attention to the condition: $>$ or $<$ And we will find the results.

The algebraic method

To solve the absolute value inequality using the algebraic method, we will divide it into two cases.
In the first case, we will consider that the expression inside the absolute value is positive and greater than zero.
In the second case, we will consider that the expression inside the absolute value is negative, less than zero.

Then, we will return to the original inequality and divide it into two cases as well.
In the first case, we can simply remove the absolute value (since it is a positive expression) and solve the inequality in the usual way.
In the second case, we will have to remove the minus sign from outside the expression to get rid of the absolute value, open parentheses, and then solve the inequality.

We will find for each case the values that also meet the conditions - the common domain for each case.
To do this, we will draw a number line and mark the fields accordingly.

Now we find all the values of $X$ that have the common domains, all the values that are in this domain satisfy the inequality.

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Inequality

Example to solve inequalities:
$3x+2<4x-13$
We will move the section:
$-x<-15$
Divide by 1 both sections and remember to convert the inequality sign as follows:
$x>15$
We have reached the solution of the inequality. We find all the values of $X$ that satisfy the inequality.
The meaning of the result is that when we set any $X$ greater than $15$, the inequality will exist.

The geometric method

Basic rule:
$│x-a│ =$ the distance between $X$ and $a$.

To solve the absolute value inequality using the geometric method, we will need to find all the values for which the distance between them and a satisfies the condition.
How do we do that?
Let's see the solution through an example:
$│x-3│<7$
We are asked, what are the values of $X$, whose distance between them is $3$, less than $7$.
We draw a number line, and mark the point of interest for the distance to $X$. In this example $-3$.
Then we will find the points whose distance from the relevant point (in this example $-3$) is exactly the distance mentioned in the condition (in this example the distance is $7$) and mark it in a different color.

Now, let's pay attention to the condition: $>$ OR $<$ . In this example, we need to find the points whose distance from $3$ is less than $7$.
That is, all the values between $10$ and $-4$.
We will mark them on the figure and the result will be: $-4
If the inequality were the same only with $7>$, the result is
$x>10$ or $x<-4$
Since these are the values whose distance from $3$ is greater than $7$.

Do you know what the answer is?

The algebraic method

We will take the same inequality that we solved geometrically and now solve it algebraically:
To solve the absolute value inequality using the algebraic method, we will divide it into two cases.
$│x-3│<7$
In the first case, we will consider that the expression inside the absolute value is positive and greater than zero.
In the second case, we will consider that the expression inside the absolute value is negative, less than zero.
We will solve the inequalities we have obtained.

First case:
$X-3\ \geq 0$
That is
$X\ \geq3$

Second case:
$X-3<0$
That is
$X<3$

Then, we will return to the original inequality and divide it into two cases as well.
In the first case, we can simply remove the absolute value (since it is a positive expression) and solve the inequality in the usual way.
In the second case, we will have to add a minus outside the expression to get rid of the absolute value, open parentheses, and then solve the inequality.
We will mark the final solutions with a different color.

Continuing with the first case:
$X-3<7$
$X<10$

Continuing with the second case:
$-(X-3)<7$
$-X+3<7$
$-X<4$
$X<-4$

Pay attention! We have not finished the solution yet.
To continue, we will find in each case the values that also meet the conditions: the common domain for each case.
To do this, we will draw a number line and mark the domains accordingly when a solid point includes the number and an open point does not include the number.
After marking the domains, we find the common domain for both domains and mark it with a different color.

Common domain:
$-4

Common domain:
$3\ \leq X>10$

And again... we are not done yet.
Now we will find all the values of $X$ that contain the common domain (yellow) of the first case or the common domain (yellow) of the second case.

We will draw the number line again and it can be easily seen:

We can see that this solution to the inequality is $-4
All values within this range maintain the inequality.
Note that we obtained the same result both algebraically and geometrically.
Of course, no matter which method you choose, if you follow the steps and do not make mistakes, you will arrive at the correct answer one way or another.

Examples and exercises with solutions of inequalities (with absolute value)

Exercise #1

Solve the inequality:

5-3x>-10

Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

5 > x

Exercise #2

Solve the following inequality:

5x+8<9

Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

x<\frac{1}{5}

Exercise #3

What is the solution to the following inequality?

$10x-4≤-3x-8$

Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

$10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

Exercise #4

Which diagram represents the solution to the inequality below? 5-8x<7x+3

Step-by-Step Solution

First, we will move the elements:

5-8x>7x+3

5-3>7x+8x
2>13x

We divide the answer by 13, and we get:

x > \frac{2}{13}

Exercise #5

Solve the inequality:

8x+a < 3x-4

Step-by-Step Solution

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!