Problem 61 - Algebra 1: Concepts and Skills

Question

Solve the absolute-value inequality. (Lesson 6.7) $$|x+5|>17$$

Step-by-Step Solution

Answer

The solution to the inequality \(|x+5| > 17\) is \(x < -22\) or \(x > 12\).

Step 1: Consider the positive aspect

The first possibility is when the quantity inside the absolute value is positive. This gives us the inequality \(x+5 > 17\). To solve for \(x\), we subtract 5 from both sides, which gives us \(x > 12\).

Step 2: Consider the negative aspect

The second possibility is when the quantity inside the absolute value is negative. Hence, this leads to the inequality \(-(x+5) > 17\), which simplifies to \(-x - 5 > 17\). Adding 5 to both sides of the equation gives us \(-x > 22\). To isolate \(x\), we multiply every term by -1, which changes the direction of the inequality, resulting in \(x < -22\).

Step 3: Combine the solution sets

Combining both solutions from Step 1 and Step 2, the solution set is \(x < -22\) or \(x > 12\).

Key Concepts

Solving Inequalities

Grasping the concept of solving inequalities is crucial for understanding absolute value problems. An inequality, unlike an equation, indicates that one side is either greater than, less than, greater than or equal to, or less than or equal to the other side. In algebra, when dealing with inequalities, we perform operations similar to those in equation solving, but with one vital difference: if we multiply or divide by a negative number, we must reverse the direction of the inequality sign.

For example, if we have the inequality \( -x > 22 \), and we wish to solve for \( x \), we cannot simply divide by \(-1\) without changing the '>' sign to '<'. This results in the inequality \( x < -22 \). Understanding this principle is key to correctly solving absolute value inequalities.

Algebraic Expressions

In algebra, expressions are combinations of numbers, variables, and arithmetic operations. In the context of absolute value problems, the expression inside the absolute value signs can be considered as a single entity, although it may contain several terms.

An expression like \(x + 5\) is one such example, which can take on different values based on the value of \(x\). When dealing with such expressions in inequalities, you need to consider both the case where the expression is positive and where it is negative, because the absolute value is the distance from zero and is always non-negative. Hence the reason you break the absolute value inequality into two separate cases to find the complete solution set.

Inequality Solution Sets

A solution set of an inequality represents all the values that satisfy the inequality. When you solve an absolute-value inequality, like \( |x+5| > 17 \), the solution set is often not just a single range of numbers, but two separate intervals. This occurs because you're considering both the positive and negative scenarios inside the absolute value.

From our example, the solution sets from both scenarios are \( x > 12 \) and \( x < -22 \), which means that any number greater than 12 or less than -22 will satisfy the original inequality. Here, the use of 'or' is essential, as we are combining two distinct sets of values. Remember, each inequality describes an interval on the number line.

Equation Solving

Equation solving is a foundational skill in algebra. It involves finding the value(s) of the variable(s) that make the equation true. However, when you're solving an inequality and not an equation, the goal is to find not just one solution, but all possible solutions that make the inequality true.

Keep in mind that while equality means both sides are exactly the same, inequality denotes a range of possibilities, excluding equality. When you transition from solving equations to solving inequalities, the methods remain familiar, but the outcome is a set of solutions instead of a single value. For instance, in the case of absolute-value inequalities, after breaking down the inequality into two separate cases based on the sign, you solve each as you would an equation to determine the ranges that constitute the solution set.