Problem 93 - College Algebra

Question

Solve absolute value inequality. \(4+\left|3-\frac{x}{3}\right| \geq 9\)

Step-by-Step Solution

Answer

There is no solution to this absolute value inequality.

Step 1: Isolate the Absolute Value

Subtract 4 from both sides to isolate the absolute value function: \((3-\frac{x}{3}) \geq 5\)

Step 2: Positivity Condition

Now consider the condition when the expression is positive or equal to zero: \(3-\frac{x}{3} \geq 5\). Solve this for x, and express the inequality in terms of x, which gives \(x \leq -6\)

Step 3: Negativity Condition

Now consider the condition when the expression is negative or equal to zero: \(3-\frac{x}{3} \leq -5\). Solve this for x, to get \(x \geq 24\)

Step 4: Final Solution

Now, since the result has to satisfy both conditions, we combine the solutions. But there's no overlap between \(x \leq -6\) and \(x \geq 24\), so there's no common solution. Hence, there's no solution to this inequality.

Key Concepts

Understanding Algebraic Inequalities

Algebraic inequalities are expressions that show the relationship between two quantities using inequality symbols like \(<, \leq, >, \geq\). These inequalities help us understand how one quantity compares to another, and they appear frequently in math problems.

• They tell us which values satisfy the given condition – for instance, \(x > 3\) means any number greater than 3 can be a solution.
• When solving inequalities, we use specific techniques similar to solving equations, but we also need to remember that the direction of the inequality symbol may change.
• Absolute value inequalities involve further considerations because the expression inside the absolute value can be positive or negative.

Mastering these inequalities is a crucial skill in algebra that lays the foundation for more advanced mathematical topics.

Isolation of Absolute Value

To solve an absolute value inequality, the first step is to isolate the absolute value expression on one side of the inequality. Doing this simplifies the problem and prepares it for further steps.

• For our problem, we start by subtracting 4 from both sides of the inequality to isolate \(|3 - \frac{x}{3}|\).
• This leads us to the simplified inequality \(|3 - \frac{x}{3}| \geq 5\).

Isolating the absolute value is essential because it allows us to clearly see what we need to consider next – whether the expressions inside the absolute value are positive or negative. This step makes solving the inequality more straightforward.

Inequality Solving Techniques

There are specific techniques to solve inequalities, especially absolute value inequalities. After isolating the absolute value, we handle the inequality based on whether the expression inside is positive or negative.

• Positivity Condition: Assume the expression inside the absolute value is non-negative. For our example, this means solving the inequality \(3 - \frac{x}{3} \geq 5\), and we find that \(x \leq -6\).
• Negativity Condition: Now assume the expression is negative. So, we solve \(3 - \frac{x}{3} \leq -5\) and find \(x \geq 24\).

These two scenarios cover all possible cases of the absolute value expression being greater than or equal to a number. Understanding and applying these techniques allows us to correctly solve inequalities.

Recognizing No Solution Cases

In solving inequalities, you might sometimes end up with conditions that have no overlap, which means there's no value of 'x' that can satisfy the inequality. This is known as a "no solution" case.

• In our problem, after solving for both positivity and negativity conditions, the solutions \(x \leq -6\) and \(x \geq 24\) have no common numbers.
• This lack of overlap is what tells us that there's no number for \(x\) that will satisfy the original inequality \(4 + |3 - \frac{x}{3}| \geq 9\).

Recognizing a no solution case is crucial as it saves time and helps avoid incorrect solutions when solving inequalities.