To solve the inequality [tex]\( |4x - 7| > 3 \)[/tex], follow these steps:
1. Break down the absolute value inequality into two separate inequalities:
When you have [tex]\( |A| > B \)[/tex], this means [tex]\( A > B \)[/tex] or [tex]\( A < -B \)[/tex].
For [tex]\( |4x - 7| > 3 \)[/tex], we can write:
[tex]\[
4x - 7 > 3 \quad \text{or} \quad 4x - 7 < -3
\][/tex]
2. Solve the first inequality [tex]\( 4x - 7 > 3 \)[/tex]:
[tex]\[
4x - 7 > 3
\][/tex]
Add 7 to both sides:
[tex]\[
4x > 10
\][/tex]
Divide by 4:
[tex]\[
x > 2.5
\][/tex]
3. Solve the second inequality [tex]\( 4x - 7 < -3 \)[/tex]:
[tex]\[
4x - 7 < -3
\][/tex]
Add 7 to both sides:
[tex]\[
4x < 4
\][/tex]
Divide by 4:
[tex]\[
x < 1
\][/tex]
4. Combine the solutions:
The solutions to the inequality [tex]\( |4x - 7| > 3 \)[/tex] are:
[tex]\[
x < 1 \quad \text{or} \quad x > 2.5
\][/tex]
5. Determine the unknown value in the provided inequality [tex]\( x < 1 \text{ or } x > \frac{5}{[?]} \)[/tex]:
From our solution, we know that the values of [tex]\( x \)[/tex] must satisfy [tex]\( x > 2.5 \)[/tex]. This means that the term [tex]\( \frac{5}{[?]} \)[/tex] must equal 2.5.
To find [tex]\( [?] \)[/tex]:
[tex]\[
\frac{5}{[?]} = 2.5
\][/tex]
Solving for [tex]\( [?] \)[/tex]:
[tex]\[
[?] = \frac{5}{2.5}
\][/tex]
Simplifying:
[tex]\[
[?] = 2
\][/tex]
Therefore, the unknown value is [tex]\( 2 \)[/tex], and the complete detailed solution is:
The inequality [tex]\( |4x - 7| > 3 \)[/tex] results in [tex]\( x < 1 \text{ or } x > \frac{5}{2} \)[/tex].
