Question

Solve the inequality |x - 1| + 9 < 13.

Graph the solution on a number line and write the solution in interval notation.

Answer 1

To solve the inequality |x - 1| + 9 < 13, subtract 9 to isolate the absolute value, then consider two cases for the inequality. The solution is -3 < x < 5, graphed as an open interval on a number line, and presented in interval notation as (-3, 5).

Step-by-step explanation:

To solve the inequality |x - 1| + 9 < 13, we first isolate the absolute value expression by subtracting 9 from both sides:

|x - 1| < 4

Next, we consider two cases because the value inside the absolute value can be either positive or negative:

1. x - 1 < 4 which simplifies to x < 5
2. -(x - 1) < 4 which simplifies to x > -3

Combining these two inequalities gives us the solution -3 < x < 5. This means that x must be greater than -3 but less than 5.

Next, we graph this on a number line by drawing an open circle at -3 and at 5, connecting them with a line to indicate all the numbers between -3 and 5 are included in the solution set but not the endpoints themselves.

Finally, we express the solution in interval notation as (-3, 5).