Sure, let's solve each inequality step by step.
### Solving the first inequality: [tex]\(-6 + x \leq 8\)[/tex]
1. Start by isolating [tex]\(x\)[/tex]:
[tex]\[
-6 + x \leq 8
\][/tex]
2. Add 6 to both sides of the inequality to solve for [tex]\(x\)[/tex]:
[tex]\[
x \leq 8 + 6
\][/tex]
3. Simplify the right side:
[tex]\[
x \leq 14
\][/tex]
So, the solution to the first inequality is:
[tex]\[
x \leq 14
\][/tex]
### Solving the second inequality: [tex]\(3(-3x + 1) < 2x - (x - 1)\)[/tex]
1. Start by distributing and simplifying both sides of the inequality:
[tex]\[
3(-3x + 1) < 2x - (x - 1)
\][/tex]
2. Distribute 3 on the left side:
[tex]\[
3 \cdot (-3x) + 3 \cdot 1 < 2x - (x - 1)
\][/tex]
[tex]\[
-9x + 3 < 2x - x + 1
\][/tex]
3. Simplify the right side:
[tex]\[
-9x + 3 < x + 1
\][/tex]
4. Combine like terms to isolate [tex]\(x\)[/tex]:
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
-9x - x + 3 < 1
\][/tex]
[tex]\[
-10x + 3 < 1
\][/tex]
5. Subtract 3 from both sides:
[tex]\[
-10x < 1 - 3
\][/tex]
[tex]\[
-10x < -2
\][/tex]
6. Divide by -10 and reverse the inequality sign (because we are dividing by a negative number):
[tex]\[
x > -\frac{2}{-10}
\][/tex]
[tex]\[
x > \frac{1}{5}
\][/tex]
So, the solution to the second inequality is:
[tex]\[
x > \frac{1}{5}
\][/tex]
### Combining both solutions:
From the first inequality, [tex]\(x \leq 14\)[/tex] and from the second inequality, [tex]\(x > \frac{1}{5}\)[/tex].
So, combining these, we get:
[tex]\[
\frac{1}{5} < x \leq 14
\][/tex]
### Graphing the Solution:
1. Draw a number line.
2. Plot an open circle at [tex]\(x = \frac{1}{5}\)[/tex] to indicate that this value is not included in the solution.
3. Plot a closed circle at [tex]\(x = 14\)[/tex] to indicate that this value is included in the solution.
4. Shade the region between [tex]\(\frac{1}{5}\)[/tex] and 14.
This represents all the values of [tex]\(x\)[/tex] that satisfy both inequalities.
So, the combined solution and its graphical representation on the number line are:
[tex]\[
\frac{1}{5} < x \leq 14
\][/tex]
