Problem 51 - Algebra for College Students

Question

Solve each equation and inequality. \(|x+7|-3 \geq 4\)

Step-by-Step Solution

Answer

The solution is \(x \geq 0\) or \(x \leq -14\).

Step 1: Isolate the Absolute Value

Begin by adding 3 to both sides of the inequality to isolate the absolute value expression: \(|x + 7| - 3 + 3 \geq 4 + 3\). This simplifies to \(|x + 7| \geq 7\).

Step 2: Set Up the Compound Inequalities

The expression \(|x + 7| \geq 7\) results in two separate inequalities: 1. \(x + 7 \geq 7\) 2. \(x + 7 \leq -7\)

Step 3: Solve the First Inequality

For \(x + 7 \geq 7\), subtract 7 from both sides to solve for \(x\): \(x + 7 - 7 \geq 7 - 7\)which simplifies to \(x \geq 0\).

Step 4: Solve the Second Inequality

For \(x + 7 \leq -7\), subtract 7 from both sides to solve for \(x\): \(x + 7 - 7 \leq -7 - 7\)which simplifies to \(x \leq -14\).

Step 5: Combine the Solutions

The solution to the inequality \(|x + 7| \geq 7\) is the union of the two solutions obtained in Steps 3 and 4. Therefore, the solution is: \(x \geq 0\) or \(x \leq -14\).

Key Concepts

Inequalities

Understanding inequalities is a fundamental skill in math. They are mathematical statements that compare two expressions. Unlike equations, which state that two expressions are equal, inequalities show whether one expression is larger or smaller than another.
You can recognize them by symbols like:

• Greater than (>).
• Less than (<).
• Greater than or equal to (≥).
• Less than or equal to (≤).

Inequalities require careful manipulation to isolate variables and find solutions. For example, if you start with an inequality like \(|x+7| - 3 \geq 4\), the goal is to isolate the variable \(x\). You want the inequality statement to focus on \(x\) itself, making it easier to understand and solve.In this case, isolating involves rearranging terms without changing the inequality's meaning. Care must be taken since performing certain operations (like multiplying or dividing by a negative number) can change the direction of the inequality.

Compound Inequalities

After isolating the absolute value expression, solving \( |x + 7| \geq 7 \) leads to compound inequalities. These involve more than one inequality statement. Absolute value inequalities involve dividing the problem into cases to consider both the positive and negative scenarios.For example, the inequality above results in two compounds:

• Case 1: \(x + 7 \geq 7\)
• Case 2: \(x + 7 \leq -7\)

Solving compound inequalities means dealing with both inequalities separately. Each represents a different potential range of solutions and, when taken together, they define the complete set of solutions.In the given exercise, after simplifying both cases, you'll arrive at solutions that cover two different intervals on the number line: \(x \geq 0\) and \(x \leq -14\). Such solutions often provide a comprehensive outlook on the possible values of \(x\) that satisfy the original inequality.

Expression Simplification

Expression simplification is a crucial step in solving equations and inequalities. It involves breaking down expressions into more manageable parts. In the given problem, you began with an absolute value inequality \( |x+7|-3 \geq 4 \). Simplifying meant removing constants from around the absolute value to focus on the core expression:Adding 3 to both sides simplified the inequality to \( |x+7| \geq 7 \). This made it easier to define the two cases for absolute value inequality discussed earlier.Simplifying expressions step-by-step not only clarifies which operations are necessary but also lays the groundwork for applying further algebraic techniques. Accurate simplification ensures that the expression is correctly interpreted without any mistakes, leading to a valid solution. Always break down each part methodically, and check your work to ensure that every step follows logically from the previous.