Solving Inequalities Involving Absolute Value
This is an inequality involving the absolute value of a variable \( d \). To solve for \( d \), we need to consider two cases because the absolute value of a number represents its distance from 0 on the number line, regardless of direction.
The inequality is:
\[ 3|d| + 5 < 47 \]
First, we will isolate the absolute value on one side by subtracting 5 from both sides:
\[ 3|d| < 42 \]
Next, we will divide both sides by 3 to solve for the absolute value of \( d \):
\[ |d| < 14 \]
Now let's consider the two cases for \( d \):
1. If \( d \) is non-negative (\( d \geq 0 \)), the absolute value sign can be removed without changing the sign of \( d \):
\[ d < 14 \]
2. If \( d \) is negative (\( d < 0 \)), we remove the absolute value sign and reverse the inequality sign because the absolute value of a negative number is positive:
\[ -d < 14 \]
\[ d > -14 \]
So the solution for the inequality \( 3|d| + 5 < 47 \) is:
\[ -14 < d < 14 \]
This means that \( d \) must be greater than -14 and less than 14.
