Problem 70 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x+3| \geq 4$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the inequality without absolute value bars

An absolute value inequality \(|x+a| \geq b\) can be rewritten as \(x + a \geq b\) or \(x + a \leq -b\). Applying this to our exercise, we get two separate inequalities \(x + 3 \geq 4\) and \(x + 3 \leq -4\).

Step 2: Solve each inequality separately

To solve the first inequality \(x + 3 \geq 4\), we subtract 3 from both sides to find \(x \geq 1\). To solve the second inequality \(x + 3 \leq -4\), we subtract 3 from both sides to find \(x \leq -7\).

Step 3: Graph the solution

Draw a real number line. Here, we shall illustrate two segments. The first one is from 1 (inclusive) to positive infinity, accompanied by a arrow depicting continuity till infinity. The other segment is from -7 (inclusive) to negative infinity. Thus demonstrating that the value of 'x' can lie in either of these two segments.

Step 4: Express the solution in interval notation

The solution is expressed in interval notation as \([-7, -\infty) \cup [1, \infty)\].

Key Concepts

Interval Notation

Interval notation is a concise way of describing a set of numbers along a number line. It uses brackets and parentheses to convey inclusivity and exclusivity of endpoints. To express a solution set in interval notation, we use the following rules:

• If the endpoint number is included in the set, we use square brackets [ ].
• If the endpoint is not included, we use round brackets ( ).

In the given problem, the solution is \([-7, -\infty) \cup [1, \infty)\). Here, the square bracket on -7 means -7 is part of the solution set, while the parenthesis with infinity signifies that the set extends indefinitely. The union symbol \(\cup\) indicates that the values of \(x\) can be in either of these intervals.

Number Line Graphing

Graphing solutions on a number line is a visual way to represent the range of solutions. To graph the solution set for an inequality, follow these steps:

• Draw a horizontal line and mark numbers on it, like -7 and 1, according to the problem's context.
• Use solid dots or closed circles to denote endpoints included in the solution set. For example, at points -7 and 1.
• Draw an arrow extending into infinity to show continuity. In this example, there would be one arrow pointing left from -7 and another pointing right from 1.

Number line graphing helps in visually understanding how the solutions spread across different segments.

Inequality Solving

Solving inequalities involves finding the values of the variable that satisfy the given condition. For absolute value inequalities, like |x + 3| \geq 4, we rewrite to remove the absolute value by creating two separate inequalities:

• First inequality: \(x + 3 \geq 4\)
• Second inequality: \(x + 3 \leq -4\)

Solving these inequalities leads to \(x \geq 1\) for the first and \(x \leq -7\) for the second. The solutions are intervals that don't overlap, showing distinct ranges for \(x\). This process ensures all possibilities of \(x\) meeting the original condition are explored.

Equivalent Inequalities

Equivalent inequalities are different expressions that represent the same solutions in terms of the set of numbers that satisfy them. When rewriting absolute value inequalities such as |x + 3| \geq 4, we establish two new inequalities:

• \(x + 3 \geq 4\) representing values above a certain point.
• \(x + 3 \leq -4\) representing values below a certain point.

These two inequalities describe the same original absolute value inequality, ensuring the solutions cover all valid scenarios. The concept of equivalence is crucial as it allows us to handle complex inequalities in a simpler form while still accurately reflecting the original problem.