Problem 62 - College Algebra

Question

Solve absolute value inequality. \(|x+3| \leq 4\)

Step-by-Step Solution

Answer

The solution to the inequality \(|x+3| \leq 4\) is \( -3 \leq x \leq 1 \).

Step 1: Split the inequality

Start by splitting the inequality into two cases. The first is when the inside of the absolute value is nonnegative \(x+3 \geq 0\), and the second is when it is negative \(x+3 < 0\). This gives two inequalities: \(x+3 \leq 4\) and \(-(x+3) \leq 4\).

Step 2: Solve the first inequality

Solve the first inequality by subtracting 3 from both sides, giving \(x \leq 1\). This is the solution when \(x+3 \geq 0\), or equivalently, when \(x \geq -3\). So the first solution is \(x \leq 1\) for \(x \geq -3\), that is, \( -3 \leq x \leq 1 \).

Step 3: Solve the second inequality

Solve the second inequality. First get rid of the minus sign by multiplying both sides by -1, which gives \(x+3 \geq -4\). Then, subtract 3 from both sides to get \(x \geq -7\). This is the solution when \(x+3 < 0\), or equivalently, when \(x < -3\). So the second solution is \(x \geq -7\) for \(x < -3\). But for \(x < -3 \), there is no solution, as it does not exist in the solution from the first part, so the second part has no solution.

Step 4: Join the solutions

As the second part does not have a solution, our solution is purely from the first part, that is, \( -3 \leq x \leq 1 \).

Key Concepts

Absolute Value Equations

Understanding absolute value equations is crucial for solving algebraic problems involving magnitude and distance. The absolute value of a number represents its distance from zero on a number line, ignoring the direction. An equation involving an absolute value expression is known as an absolute value equation.

For example, the equation
\(|x+3| \leq 4\)
involves the absolute value of the expression \(x+3\). To solve such an equation, we consider two scenarios because the expression inside the absolute value can be non-negative (resulting in itself) or negative (resulting in its opposite).

In the given exercise, we split the problem into two separate inequalities:
\(x+3 \leq 4\)
and
\(x+3 > -4\).
The solutions to these inequalities are then combined to find the range of values that satisfy the original absolute value inequality.

Inequality Solutions

Inequality solutions are integral in algebra, providing a range of values, rather than a single value as in equations. When faced with an inequality, such as
\(|x+3| \leq 4\),
we're looking for all possible values of \(x\) that make the inequality true.

The solution steps outlined demonstrate the process of solving inequalities step by step:

• Split the absolute value inequality into two separate cases based on the sign of the expression within the absolute value.
• Independently solve each inequality.
• Finally, interpret the results by considering the intersection of solutions.
• Remember, when multiplying or dividing an inequality by a negative number, the inequality symbol flips.

The solution process ensures that all possible values of \(x\) are considered, leading to a comprehensive understanding of inequality solutions.

Algebraic Inequalities

Algebraic inequalities are a fundamental concept in mathematics that express the relationship between two expressions. In the context of algebraic inequalities, we often deal with expressions involving variables, and we must find the values of these variables for which the inequality holds true.

In the exercise, we encounter an algebraic inequality with absolute values. These inequalities have special implications, as the solutions may not be a single interval but could be two or more intervals that satisfy the inequality. As we analyze the solutions in this problem, understanding the properties and rules for solving algebraic inequalities is crucial. These skills not only help students solve textbook exercises but also build a foundation for solving real-world problems where determining a range of possibilities is essential.