To solve the inequality [tex]\(6 + 13x + 2 > 2 - 13x\)[/tex], follow these steps:
Step 1: Simplify both sides of the inequality.
[tex]\[ 6 + 13x + 2 > 2 - 13x \][/tex]
Combine the constant terms on the left side:
[tex]\[ 8 + 13x > 2 - 13x \][/tex]
Step 2: Add [tex]\(13x\)[/tex] to both sides to eliminate the [tex]\(x\)[/tex] term on the right side.
[tex]\[ 8 + 13x + 13x > 2 - 13x + 13x \][/tex]
[tex]\[ 8 + 26x > 2 \][/tex]
Step 3: Subtract 8 from both sides to isolate the [tex]\(x\)[/tex] term.
[tex]\[ 8 + 26x - 8 > 2 - 8 \][/tex]
[tex]\[ 26x > -6 \][/tex]
Step 4: Divide both sides by 26 to solve for [tex]\(x\)[/tex].
[tex]\[ \frac{26x}{26} > \frac{-6}{26} \][/tex]
[tex]\[ x > -\frac{6}{26} \][/tex]
[tex]\[ x > -\frac{3}{13} \][/tex]
Hence, the solution to the inequality [tex]\(6 + 13x + 2 > 2 - 13x\)[/tex] is:
[tex]\[ x > -\frac{3}{13} \][/tex]
None of the provided options directly match [tex]\( x > -\frac{3}{13} \)[/tex]. However, the solution correctly indicates all values of [tex]\(x\)[/tex] greater than [tex]\(-\frac{3}{13}\)[/tex], or in interval notation, [tex]\( \left( -\frac{3}{13}, \infty \right) \)[/tex]. It appears there may be an error in the provided answer choices.
When considering the answer choices:
A. [tex]\( x < 2 \)[/tex]: This restricts [tex]\(x\)[/tex] to be less than 2, which is not equivalent to our solution.
B. [tex]\( x \geq -1 \)[/tex]: This includes [tex]\(-1\)[/tex], which is less strict than our solution.
C. [tex]\( x > 1 \)[/tex]: This restricts [tex]\(x\)[/tex] to be greater than 1, which is incorrect.
D. [tex]\( x < -2 \)[/tex]: This is completely incorrect as it is not in the solution range.
To further assist with assumptions or rephrased questions subsequently, please ensure the problem statement and options are aligned with the derived mathematical steps.
