Certainly! To solve the inequality [tex]\( |4x - 7| > 3 \)[/tex], we need to analyze the inequality by considering the definition of absolute value.
The absolute value inequality [tex]\( |A| > B \)[/tex] (where [tex]\( B \)[/tex] is a positive number) can be split into two separate inequalities:
[tex]\[ A > B \quad \text{or} \quad A < -B \][/tex]
In this case, [tex]\( A = 4x - 7 \)[/tex] and [tex]\( B = 3 \)[/tex]. Therefore, we split the inequality [tex]\( |4x - 7| > 3 \)[/tex] into:
[tex]\[ 4x - 7 > 3 \quad \text{or} \quad 4x - 7 < -3 \][/tex]
Now, let's solve each inequality separately.
### Solving [tex]\( 4x - 7 > 3 \)[/tex]:
1. Add 7 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
4x - 7 + 7 > 3 + 7
\][/tex]
Simplifying, we get:
[tex]\[
4x > 10
\][/tex]
2. Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x > \frac{10}{4}
\][/tex]
Simplify the fraction:
[tex]\[
x > 2.5
\][/tex]
### Solving [tex]\( 4x - 7 < -3 \)[/tex]:
1. Add 7 to both sides:
[tex]\[
4x - 7 + 7 < -3 + 7
\][/tex]
Simplifying, we get:
[tex]\[
4x < 4
\][/tex]
2. Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{4}{4}
\][/tex]
Simplify the fraction:
[tex]\[
x < 1
\][/tex]
### Combining the results
The solution to the inequality [tex]\( |4x - 7| > 3 \)[/tex] is the union of the two separate inequalities. Therefore, the complete solution is:
[tex]\[ x < 1 \quad \text{or} \quad x > 2.5 \][/tex]
So, in the inequality form:
[tex]\[ x < 1 \quad \text{or} \quad x > 2.5 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the original inequality are those where [tex]\( x \)[/tex] is less than 1 or greater than 2.5.
