Problem 85 - Algebra for College Students

Question

Solve each equation or inequality. $$ |x+5|-6 \leq-1 $$

Step-by-Step Solution

Answer

x is in the interval [-10, 0].

Step 1 - Isolate the Absolute Value

Add 6 to both sides of the inequality to isolate the absolute value expression. |x + 5| - 6 + 6 ≤ -1 + 6 |x + 5| ≤ 5

Step 2 - Define the Absolute Value Inequality

An absolute value inequality, |x + 5| ≤ 5, can be rewritten as a compound inequality: -5 ≤ x + 5 ≤ 5.

Step 3 - Solve the Compound Inequality

Subtract 5 from each part of the compound inequality to solve for x: -5 - 5 ≤ x + 5 - 5 ≤ 5 - 5 -10 ≤ x ≤ 0

Step 4 - Write the Solution

The solution to the inequality is all values of x in the interval [-10, 0].

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute values, like \(|x + 5| ≤ 5\). Absolute value represents the distance of a number from zero on a number line, no matter the direction. For example, the absolute value of -3 is 3.

When solving absolute value inequalities, the goal is to isolate the absolute value expression and rewrite it without the absolute value. This often divides into two separate inequalities, known as compound inequalities. Knowing the definition and properties of absolute value is key. Here, \(|x + 5| ≤ 5\) means x + 5 is between -5 and 5.

Compound Inequalities

Compound inequalities are formed when two inequalities are combined. They show a range of possible values for the variable. For example, \(-5 ≤ x + 5 ≤ 5\) is a compound inequality. This is formed when you solve the absolute value inequality \(|x + 5| ≤ 5\).

To solve compound inequalities:

• Isolate the absolute value first.
• Split into two inequalities: one representing the positive and one the negative scenario.
• Solve each part separately.

Here’s how it breaks down:

• -5 ≤ x + 5 represents the lower boundary.
• x + 5 ≤ 5 represents the upper boundary.

Solving each part, we subtract 5 from all parts of \(-5 ≤ x + 5 ≤ 5\) to get \(-10 ≤ x ≤ 0\). This means x is between -10 and 0.

Interval Notation

Interval notation is a concise way of representing a range of values, which is particularly useful when dealing with inequalities. For example, the solution to the inequality \(-10 ≤ x ≤ 0\) can be written in interval notation as \([-10, 0]\).

Here are some key points about interval notation:

• [ ] include the endpoint.
• ( ) exclude the endpoint.
• Commas separate the endpoints of the interval.

So, in \([-10, 0]\), the square brackets indicate that -10 and 0 are included in the solution set. This notation simplifies understanding and communicating the range of x values that satisfy the inequality.