To solve the equation [tex]\( |5x + 30| = 15 \)[/tex], we need to consider the definition of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore,
[tex]\[ |5x + 30| = 15 \][/tex]
implies two separate equations:
[tex]\[ 5x + 30 = 15 \][/tex]
and
[tex]\[ 5x + 30 = -15 \][/tex]
Let's solve these equations step by step.
### First Equation: [tex]\( 5x + 30 = 15 \)[/tex]
1. Subtract 30 from both sides:
[tex]\[ 5x + 30 - 30 = 15 - 30 \][/tex]
[tex]\[ 5x = -15 \][/tex]
2. Divide both sides by 5:
[tex]\[ x = \frac{-15}{5} \][/tex]
[tex]\[ x = -3 \][/tex]
### Second Equation: [tex]\( 5x + 30 = -15 \)[/tex]
1. Subtract 30 from both sides:
[tex]\[ 5x + 30 - 30 = -15 - 30 \][/tex]
[tex]\[ 5x = -45 \][/tex]
2. Divide both sides by 5:
[tex]\[ x = \frac{-45}{5} \][/tex]
[tex]\[ x = -9 \][/tex]
So, the solutions to the equation [tex]\( |5x + 30| = 15 \)[/tex] are:
[tex]\[ x = -3 \quad \text{or} \quad x = -9 \][/tex]
### Graph the solutions on the number line
You should place points at [tex]\( x = -3 \)[/tex] and [tex]\( x = -9 \)[/tex] on the number line:
- A point at [tex]\( x = -3 \)[/tex]
- Another point at [tex]\( x = -9 \)[/tex]
The final solutions are:
[tex]\[ x = -3 \quad \text{or} \quad x = -9 \][/tex]
