Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x+2|+0.1 \geq 0.2 $$

Step-by-Step Solution

Answer

The solution is \[ x \in (-\infty, -2.1] \cup [-1.9, \infty) \].

Step 1: Understand the Inequality

The inequality to solve is \(|x + 2| + 0.1 \geq 0.2\). We need to find all values of \(x\) such that this inequality holds true.

Step 2: Isolate the Absolute Value

Subtract 0.1 from both sides of the inequality: \[ |x + 2| + 0.1 - 0.1 \geq 0.2 - 0.1 \]This simplifies to:\[ |x + 2| \geq 0.1 \]

Step 3: Solve the Absolute Value Inequality

The inequality \(|x + 2| \geq 0.1\) implies two scenarios:1. \(x + 2 \geq 0.1\)2. \(x + 2 \leq -0.1\)For scenario 1:\[ x + 2 \geq 0.1 \]Subtract 2 from both sides:\[ x \geq 0.1 - 2 \]\[ x \geq -1.9 \]For scenario 2:\[ x + 2 \leq -0.1 \]Subtract 2 from both sides:\[ x \leq -0.1 - 2 \]\[ x \leq -2.1 \]

Step 4: Express the Solution in Interval Notation

Combine the intervals from the two scenarios. The solution to the inequality is:\[ x \in (-\infty, -2.1] \cup [-1.9, \infty) \]This means \(x\) can be any number less than or equal to -2.1 or greater than or equal to -1.9.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always a non-negative value, which means it can't be less than zero. When we see an expression like \(|x+2|\), we're looking at how far the value \(x+2\) is from zero.

• For example, \(|3| = 3\) because 3 is three units away from zero.
• Similarly, \(|-3| = 3\) because -3 is also three units away, just in the opposite direction.

When solving absolute value inequalities like \(|x+2| \geq 0.1\), it means we need the distance of \(x+2\) from zero to be no less than 0.1. Therefore, we explore two possibilities: the expression \(x+2\) is greater than or equal to 0.1, or less than or equal to -0.1. This results from how absolute value addresses both the positive and negative directions.

Expressing Solutions with Interval Notation

Interval notation is a mathematical shorthand used to express ranges of values that satisfy an inequality. It is compact and provides a quick way of describing sets of numbers.
Interval notation uses brackets and parentheses to show which endpoints are included or excluded:

• Parentheses, \(( \text{ and } )\), are used when the endpoint is not included (open interval).
• Brackets, \([ \text{ and } ]\), are used when the endpoint is included (closed interval).

For the solution of the inequality \(|x+2| \geq 0.1\), we express this in interval notation as \((-\infty, -2.1] \cup [-1.9, \infty)\). This tells us:

• A range from negative infinity up to and including -2.1.
• Another range starting from -1.9 and going to positive infinity.

This notation clearly conveys that any \(x\) within these intervals satisfies our original inequality.

Steps in Solving Inequalities

Solving inequalities involves finding the set of values that make the inequality true. Let's break it down using the given inequality \(|x+2| + 0.1 \geq 0.2\):
First, simplify the inequality by isolating the absolute value:

• Subtract 0.1 from both sides: \(|x+2| \geq 0.1\).

Next, interpret the absolute value inequality by considering two possible scenarios:

• \(x+2 \geq 0.1\) leading to \(x \geq -1.9\).
• \(x+2 \leq -0.1\) leading to \(x \leq -2.1\).

Finally, express the solution using interval notation, combining the solutions from both scenarios as \((-\infty, -2.1] \cup [-1.9, \infty)\).
These steps help solve inequalities in a structured way, ensuring that we account for all possible values of \(x\) that make the inequality true.