To solve the equation [tex]\( |a + 9| = 2 \)[/tex], we need to consider the definition of the absolute value. The absolute value of [tex]\( a + 9 \)[/tex] equals 2 means that the expression [tex]\( a + 9 \)[/tex] can either be 2 or -2.
Let's break this down into two separate cases:
### Case 1: [tex]\( a + 9 = 2 \)[/tex]
1. Solve for [tex]\( a \)[/tex].
[tex]\[
a + 9 = 2
\][/tex]
2. Subtract 9 from both sides to isolate [tex]\( a \)[/tex].
[tex]\[
a = 2 - 9
\][/tex]
3. Simplify the result.
[tex]\[
a = -7
\][/tex]
### Case 2: [tex]\( a + 9 = -2 \)[/tex]
1. Solve for [tex]\( a \)[/tex].
[tex]\[
a + 9 = -2
\][/tex]
2. Subtract 9 from both sides to isolate [tex]\( a \)[/tex].
[tex]\[
a = -2 - 9
\][/tex]
3. Simplify the result.
[tex]\[
a = -11
\][/tex]
Thus, the solution set comprises the values [tex]\( a = -7 \)[/tex] and [tex]\( a = -11 \)[/tex].
### Solution Set on a Number Line
To represent these solutions on a number line:
1. Draw a horizontal line and mark some points for reference.
2. Identify and clearly mark the points [tex]\( -7 \)[/tex] and [tex]\( -11 \)[/tex] on this line.
[tex]\[
\begin{array}{*{20}{c}}
\text{} & \text{} & \text{} & \text{} & \text{-11} & \text{} & \text{} & \text{} & \text{} & \text{-7} & \text{} \\
\vdots \text{-12} & \text{} & \text{} & \text{} & \text{\bullet} & \text{} & \text{} & \text{} & \text{} & \text{\bullet} & \vdots \text{-6}
\end{array}
\][/tex]
Therefore, the solutions to the equation [tex]\( |a + 9| = 2 \)[/tex] are [tex]\( a = -7 \)[/tex] and [tex]\( a = -11 \)[/tex], and they are represented as points on the number line.
