Let's solve the given equation step-by-step to find the smaller and larger values of [tex]\( x \)[/tex].
The given equation is:
[tex]\[ (5x - 1)^2 = 48 \][/tex]
To solve for [tex]\( x \)[/tex], follow these steps:
1. Take the square root of both sides of the equation:
[tex]\[ 5x - 1 = \pm \sqrt{48} \][/tex]
2. Calculate the value of [tex]\(\sqrt{48}\)[/tex]:
[tex]\[ \sqrt{48} \approx 6.928203230275509 \][/tex]
Now, we have two separate equations to solve:
First equation:
[tex]\[ 5x - 1 = \sqrt{48} \][/tex]
Second equation:
[tex]\[ 5x - 1 = -\sqrt{48} \][/tex]
3. Solve each equation for [tex]\( x \)[/tex]:
For the first equation:
[tex]\[ 5x - 1 = 6.928203230275509 \][/tex]
Add 1 to both sides:
[tex]\[ 5x = 6.928203230275509 + 1 \][/tex]
[tex]\[ 5x = 7.928203230275509 \][/tex]
Divide by 5:
[tex]\[ x = \frac{7.928203230275509}{5} \][/tex]
[tex]\[ x \approx 1.5856406460551018 \][/tex]
For the second equation:
[tex]\[ 5x - 1 = -6.928203230275509 \][/tex]
Add 1 to both sides:
[tex]\[ 5x = -6.928203230275509 + 1 \][/tex]
[tex]\[ 5x = -5.928203230275509 \][/tex]
Divide by 5:
[tex]\[ x = \frac{-5.928203230275509}{5} \][/tex]
[tex]\[ x \approx -1.1856406460551017 \][/tex]
4. Determine the smaller and larger values of [tex]\( x \)[/tex]:
- The smaller value is:
[tex]\[ x \approx -1.1856406460551017 \][/tex]
- The larger value is:
[tex]\[ x \approx 1.5856406460551018 \][/tex]
Therefore, the solution is:
Smaller value:
[tex]\[ x = -1.1856406460551017 \][/tex]
Larger value:
[tex]\[ x = 1.5856406460551018 \][/tex]
