Suppose that |x-4|<1 (a) What is the maximum possible value of |x+4| ? (b) Show that |

step 1 In this exercise, the first thing that we want to do is to express this in a different way. X mi

Suppose that |x-4|<1 (a) What is the maximum possible...

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Key Concepts

Absolute Value Inequalities

This concept involves understanding that an inequality such as |x - a| < b describes all numbers x within a distance b from the number a. It allows one to interpret problems in terms of distances on the number line, which in turn facilitates locating the range of x values that satisfy the condition and studying the behavior of functions defined on these intervals.

Factoring and the Difference of Squares

Recognizing that an expression like x² - c² can be factored into (x - c)(x + c) is a key algebraic concept. This factorization simplifies the analysis by breaking down a quadratic expression into a product of two linear factors, which can be individually bounded or manipulated using given inequalities.

Bounding Expressions Using Known Constraints

Once the range of a variable is established from an inequality, one can use it to find bounds on related expressions. By combining the known constraints on x with algebraic manipulations—such as multiplying bounds derived from absolute value inequalities—one can determine maximum or minimum values of other functions of x. This method is particularly useful in problems requiring the estimation of an expression’s upper or lower limit.

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