Final answer:
To solve the inequality |7x| - 3 ≥ 53, we need to consider two scenarios due to the absolute value, resulting in x ≤ -8 or x ≥ 8. These solutions represent the range of x values that satisfy the original inequality.
Step-by-step explanation:
To solve the inequality |7x| - 3 ≥ 53, we first isolate the absolute value expression by adding 3 to both sides:
|7x| ≥ 56
This results in two separate inequalities due to the nature of absolute values:
- 7x ≥ 56
- -7x ≥ 56
Solving each inequality we get:
- x ≥ 8
- x ≤ -8
The solution to the original inequality is the union of the solutions to these two inequalities, which in interval notation is x ≤ -8 or x ≥ 8.
Understanding the properties of inequalities and absolute value is critical for solving problems like this. Multiplication and division rules for positive and negative numbers, as discussed in the reference information, also play a role in simplifying and solving such inequalities.