Answer :
To convert the inequality [tex]\( x < 7 \)[/tex] into a compound inequality using the definition of absolute value, let's start by explaining the concept of absolute value and how it applies to inequalities.
### Step-by-Step Solution
1. Understand Absolute Value:
The absolute value of a number [tex]\( x \)[/tex], denoted as [tex]\( |x| \)[/tex], is defined as the non-negative value of [tex]\( x \)[/tex] without regard to its sign. Formally, this is written as:
[tex]\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \][/tex]
2. Relate to the Given Inequality:
We start with the given inequality:
[tex]\[ x < 7 \][/tex]
3. Consider Absolute Value Representation:
In this case, we aim to create an equivalent compound inequality using absolute value. For an inequality like [tex]\( x < 7 \)[/tex], we understand that [tex]\( x \)[/tex] can take any values less than 7.
4. Rewrite as Compound Inequality:
Since [tex]\( x \)[/tex] can be any value less than 7, we realize that this inequality is already in its simplest form. It straightforwardly represents all values [tex]\( x \)[/tex] that satisfy:
[tex]\[ x < 7 \][/tex]
### Graphing the Solution Set:
To graph the solution set of [tex]\( x < 7 \)[/tex] on the rectangular coordinate plane, follow these steps:
1. Identify the Boundary:
The boundary line for [tex]\( x = 7 \)[/tex] is vertical because we are only considering values of [tex]\( x \)[/tex].
2. Draw the Line:
Draw a vertical dashed line at [tex]\( x = 7 \)[/tex] because the inequality is strict (<, not ≤). A dashed line indicates that values on the line [tex]\( x = 7 \)[/tex] are not included in the solution set.
3. Shade the Solution Region:
Shade the region to the left of the vertical dashed line [tex]\( x = 7 \)[/tex], because that represents all values of [tex]\( x \)[/tex] that are less than 7.
### Final Graph:
On a rectangular coordinate plane, you will see:
- A vertical dashed line at [tex]\( x = 7 \)[/tex].
- The region to the left of this line is shaded, indicating all [tex]\( x \)[/tex] values that satisfy [tex]\( x < 7 \)[/tex].
The inequality [tex]\( x < 7 \)[/tex] is thus expressed as it stands, and graphically this inequality represents all points in a coordinate plane where the [tex]\( x \)[/tex]-coordinate is less than 7.
### Step-by-Step Solution
1. Understand Absolute Value:
The absolute value of a number [tex]\( x \)[/tex], denoted as [tex]\( |x| \)[/tex], is defined as the non-negative value of [tex]\( x \)[/tex] without regard to its sign. Formally, this is written as:
[tex]\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \][/tex]
2. Relate to the Given Inequality:
We start with the given inequality:
[tex]\[ x < 7 \][/tex]
3. Consider Absolute Value Representation:
In this case, we aim to create an equivalent compound inequality using absolute value. For an inequality like [tex]\( x < 7 \)[/tex], we understand that [tex]\( x \)[/tex] can take any values less than 7.
4. Rewrite as Compound Inequality:
Since [tex]\( x \)[/tex] can be any value less than 7, we realize that this inequality is already in its simplest form. It straightforwardly represents all values [tex]\( x \)[/tex] that satisfy:
[tex]\[ x < 7 \][/tex]
### Graphing the Solution Set:
To graph the solution set of [tex]\( x < 7 \)[/tex] on the rectangular coordinate plane, follow these steps:
1. Identify the Boundary:
The boundary line for [tex]\( x = 7 \)[/tex] is vertical because we are only considering values of [tex]\( x \)[/tex].
2. Draw the Line:
Draw a vertical dashed line at [tex]\( x = 7 \)[/tex] because the inequality is strict (<, not ≤). A dashed line indicates that values on the line [tex]\( x = 7 \)[/tex] are not included in the solution set.
3. Shade the Solution Region:
Shade the region to the left of the vertical dashed line [tex]\( x = 7 \)[/tex], because that represents all values of [tex]\( x \)[/tex] that are less than 7.
### Final Graph:
On a rectangular coordinate plane, you will see:
- A vertical dashed line at [tex]\( x = 7 \)[/tex].
- The region to the left of this line is shaded, indicating all [tex]\( x \)[/tex] values that satisfy [tex]\( x < 7 \)[/tex].
The inequality [tex]\( x < 7 \)[/tex] is thus expressed as it stands, and graphically this inequality represents all points in a coordinate plane where the [tex]\( x \)[/tex]-coordinate is less than 7.
