Let's solve the inequality [tex]\(9 + x < 5\)[/tex] step by step.
1. Isolate [tex]\(x\)[/tex]:
We need to isolate the variable [tex]\(x\)[/tex] on one side of the inequality. To do this, we'll subtract 9 from both sides of the inequality.
[tex]\[
9 + x < 5
\][/tex]
Subtract 9 from both sides:
[tex]\[
x < 5 - 9
\][/tex]
Simplify the right side:
[tex]\[
x < -4
\][/tex]
2. Interpret the Solution:
The solution to the inequality is all values of [tex]\(x\)[/tex] that are less than [tex]\(-4\)[/tex].
3. Plot the Solution on the Number Line:
- Mark the point [tex]\(-4\)[/tex] on the number line.
- Since [tex]\(x\)[/tex] is less than [tex]\(-4\)[/tex] and not equal to it, we use an open circle (or open endpoint) at [tex]\(-4\)[/tex] to indicate that [tex]\(-4\)[/tex] is not included in the solution set.
- Draw an arrow extending to the left from the open circle to show all values less than [tex]\(-4\)[/tex].
Here is the number line with the solution plotted:
[tex]\[
\begin{array}{ccccccccccc}
-10 & & -5 & & 0 & & 5 & & 10 \\
| & & | & & | & & | & & | \\
\end{array}
\][/tex]
On this number line, the open circle at [tex]\(-4\)[/tex] and the arrow extending to the left represent all values of [tex]\(x\)[/tex] that satisfy the inequality [tex]\(x < -4\)[/tex]:
[tex]\[
\begin{array}{cccccc|ccccc}
-10 & -9 & -8 & -7 & -6 & -5 & -4 & -3 & -2 & -1 & 0 \\
\\
\leftarrow \bullet & ) & ) & ) & ) & ) & O & & & & \\
\end{array}
\][/tex]
The open circle at [tex]\(-4\)[/tex] shows that the value [tex]\(-4\)[/tex] is not included in the solution, and the arrow pointing to the left indicates that all values less than [tex]\(-4\)[/tex] are part of the solution set.
