Problem 57 - Algebra 1: Concepts and Skills

Question

Solve the absolute-value inequality. (Lesson 6.7) $$|x+7|<12$$

Step-by-Step Solution

Answer

'x' lies in the interval (-19, 5).

Step 1: Set up two independent inequalities

First, set up two separate inequalities based on the definition of absolute value. In this case the inequalities would be:\n1. \(x+7<12\)\n2. \(-(x+7)<12\) (which simplifies to \(x+7>-12\))

Step 2: Solve the inequalities

Next, the inequalities can be solved separately. For the first inequality (i.e., \(x+7<12\)), subtract 7 from both sides to get \(x<5\). Similarly, for the second inequality (i.e., \(x+7>-12\)), subtract 7 from both sides to get \(x>-19\).

Step 3: Combined solution

Based on the two solutions obtained in step 2, the range for 'x' is obtained. So, based on this, we can state that '-19' is less than 'x' which in turn is less than '5'. In other words, 'x' lies in the range (-19, 5).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. In the algebraic context, variables such as 'x' are used to represent unknown quantities. Algebraic operations like addition, subtraction, multiplication, and division can be applied to these variables just as they are to numbers.

Understanding algebra is crucial because it serves as the foundation for more complex fields of mathematics and science. In our absolute value inequality problem, algebra comes into play as we work with variables and operations to find the range of possible values that satisfy the given inequality.

Inequality Solving

Solving inequalities is a fundamental skill in algebra that involves finding all the possible values of a variable that make an inequality true. Unlike equations, inequalities do not indicate equality but rather a range of values that allow the inequality to hold.

When solving inequalities, similar steps as in solving equations are followed, with one critical difference – if you multiply or divide by a negative number, the inequality sign flips direction. In our problem, we treat the absolute value inequality similarly but remember, it results in two separate inequalities that must be solved.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Symbolically, it is represented by two vertical bars, like this: \( |a| \). Absolute values are always non-negative.

When dealing with inequalities that include absolute values, such as \( |x+7|<12 \), we understand it as 'the distance of \( x+7 \) from zero is less than 12'. To solve it, we need to consider both the positive and negative scenarios, leading us to set up two separate inequalities reflecting the distance in both directions from zero.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value. In our exercise \( |x+7|<12 \), \( x+7 \) is one such algebraic expression.

In the context of our inequality problem, algebraic expressions are used to represent the quantities within the absolute value symbol and are manipulated to solve for the variable. Manipulating these expressions requires careful observation of the rules of algebra to ensure the accurate solution of the inequality.