Problem 72 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |x+3| \geq 4 $$

Step-by-Step Solution

Answer

The solution to the inequality \( |x+3| \geq 4 \) is \( x \leq -7 \) or \( x \geq 1 \), or in interval notation, \( (-\infty, -7] \cup [1, \infty) \).

Step 1: Analyzing the absolute value inequality

In this step, the absolute value inequality \( |x+3| \geq 4 \) is given. It can be split into two inequalities: \( x+3 \leq -4 \) and \( x+3 \geq 4 \), according to the property of absolute values.

Step 2: Solving the inequality - Case 1

First, solve the inequality \( x+3 \leq -4 \). Subtract 3 from both sides to isolate x, giving \( x \leq -4 - 3 \), which simplifies to \( x \leq -7 \). This is the first part of the solution.

Step 3: Solving the inequality - Case 2

Next, solve the inequality \( x+3 \geq 4 \). Subtract 3 from both sides again, giving \( x \geq 4 - 3 \), which simplifies to \( x \geq 1 \). This is the second part of the solution.

Step 4: Combining the solutions

The solution to the original absolute value inequality is the union of the solutions from cases 1 and 2, i.e., \( x \leq -7 \) or \( x \geq 1 \). In interval notation, the solution can be written as \( (-\infty, -7] \cup [1, \infty) \).

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is a fundamental skill in algebra. Inequalities resemble equations but instead of an equal sign, they use signs that denote less than, greater than, or their inclusive counterparts. When solving an inequality, the goal is to determine all the values of the variable that make the inequality true.

To do this, similar steps to solving equations are used, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. However, a crucial difference is when multiplying or dividing by a negative number, the inequality sign must be flipped. For example, if you have an inequality like \( -2x > 6 \), dividing by -2 to solve for \( x \) requires reversing the inequality to \( x < -3 \).

Solving absolute value inequalities involves creating two separate inequalities, as seen with \( |x+3| \geq 4 \) in the exercise. This separation accounts for the nature of absolute values which can represent both positive and negative distances from zero.

Absolute Value Properties

The absolute value of a number is its distance from zero on the number line, regardless of the direction. This distance is always non-negative. The absolute value of \( x \) is denoted as \( |x| \) and it has two primary properties:

• \( |x| = x \) if \( x \geq 0 \), meaning if the value inside the absolute value is non-negative, the absolute value is the same as the original number.
• \( |x| = -x \) if \( x < 0 \), which implies that if the value is negative, the absolute value is the positive counterpart of the number.

When solving an absolute value inequality such as \( |x+3| \geq 4 \), one must consider both cases: when \( x+3 \) is positive and when it is negative. This leads to the two inequalities \( x+3 \geq 4 \) and \( x+3 \leq -4 \), representing all the potential distances from zero.

Interval Notation

Interval notation is a way of denoting sets of numbers that fall within a specified range. This notation is compact and precise, making it a great tool for expressing solutions to inequalities. An interval is written with a starting point, an endpoint, and notation to indicate whether these points are included or excluded.

Here's how it's done:

• Brackets \( [ ] \) mean the endpoint is included, as in \( [a, b] \) includes both \( a \) and \( b \).
• Parentheses \( ( ) \) mean the endpoint is excluded, so \( (a, b) \) includes neither \( a \) nor \( b \).
• The symbols \( -\infty \) and \( \infty \) are used to signify that there is no lower or upper bound, respectively. These are always paired with parentheses since infinity is not a number that can be reached or included.

In the case of \( (-\infty, -7] \cup [1, \infty) \) from our absolute value inequality, it includes all numbers less than or equal to -7 and all numbers greater than or equal to 1, but not the numbers in between.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They become exceedingly important while solving for unknowns in algebraic problems, such as inequalities and equations. Variables stand in for unknown values, and the goal is to manipulate the expression to solve for these variables.

For instance, in our absolute value inequality \( |x+3| \geq 4 \), \( x+3 \) is an algebraic expression. When solving for \( x \) here, we consider the fact that absolute values affect the sign of the expression. Thus, we modify our approach slightly compared to straight equations: we consider both the expression as is \( (x+3) \) and its opposite \( -(x+3) \) to fully account for what the absolute value represents, eventually arriving at the dual inequalities that help solve the original problem.