Find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts of the equation:
[tex]\[ y = |x + 5| - 4 \][/tex]

Write each intercept as an ordered pair. If there is more than one intercept, separate them using "and." Select "None" if applicable.



Answer :

To find the intercepts for the given equation [tex]\( y = |x + 5| - 4 \)[/tex], we need to determine where the graph of the equation intersects the x-axis (x-intercept) and the y-axis (y-intercept).

### Finding the y-intercept:

The y-intercept occurs where the graph intersects the y-axis. This happens when [tex]\( x = 0 \)[/tex].

1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = |0 + 5| - 4 \][/tex]

2. Simplify the expression inside the absolute value:
[tex]\[ y = |5| - 4 \][/tex]

3. Evaluate the absolute value and the subtraction:
[tex]\[ y = 5 - 4 = 1 \][/tex]

Therefore, the y-intercept, written as an ordered pair, is:
[tex]\[ (0, 1) \][/tex]

### Finding the x-intercepts:

The x-intercepts occur where the graph intersects the x-axis. This happens when [tex]\( y = 0 \)[/tex].

1. Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = |x + 5| - 4 \][/tex]

2. Rearrange the equation to isolate the absolute value:
[tex]\[ 4 = |x + 5| \][/tex]

3. Solve the absolute value equation [tex]\( |x + 5| = 4 \)[/tex]. There are two possibilities:
[tex]\[ x + 5 = 4 \quad \text{or} \quad x + 5 = -4 \][/tex]

4. Solve each equation separately:
- For [tex]\( x + 5 = 4 \)[/tex]:
[tex]\[ x = 4 - 5 \][/tex]
[tex]\[ x = -1 \][/tex]

- For [tex]\( x + 5 = -4 \)[/tex]:
[tex]\[ x = -4 - 5 \][/tex]
[tex]\[ x = -9 \][/tex]

Therefore, the x-intercepts, written as ordered pairs, are:
[tex]\[ (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]

### Conclusion:

The intercepts for the equation [tex]\( y = |x + 5| - 4 \)[/tex] are:
[tex]\[ \text{y-intercept: } (0, 1) \][/tex]
[tex]\[ \text{x-intercepts: } (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]