Problem 89 - Algebra for College Students

Question

Solve each equation or inequality. $$ |0.5 x-3.5|+0.2 \geq 0.6 $$

Step-by-Step Solution

Answer

x \leq 6.2 \text{ or } \x \geq 7.8

Step 1 - Isolate the absolute value

First, isolate the absolute value expression by subtracting 0.2 from both sides of the inequality: \[|0.5x - 3.5| + 0.2 - 0.2 \geq 0.6 - 0.2 \]This simplifies to: \[|0.5x - 3.5| \geq 0.4 \]

Step 2 - Set up the compound inequality

The inequality \(|A| \geq B\) means that A is either greater than or equal to B, or A is less than or equal to -B. Set up two separate inequalities: \[0.5x - 3.5 \geq 0.4 \text{ or } \0.5x - 3.5 \leq -0.4 \]

Step 3 - Solve the first inequality

Solve the first inequality: \[0.5x - 3.5 \geq 0.4 \]Add 3.5 to both sides:\[0.5x \geq 3.9 \]Then divide by 0.5:\[ x \geq 7.8 \]

Step 4 - Solve the second inequality

Solve the second inequality: \[0.5x - 3.5 \leq -0.4 \]Add 3.5 to both sides:\[0.5x \leq 3.1 \]Then divide by 0.5:\[ x \leq 6.2 \]

Step 5 - Combine the solutions

Combine both solutions. The solutions for the inequality are: \[ x \leq 6.2 \text{ or } \x \geq 7.8 \]

Key Concepts

absolute value

The concept of absolute value refers to the magnitude of a number without considering its direction. In other words, it tells us how far a number is from zero on a number line, regardless of whether the number is positive or negative. The absolute value of a number is always non-negative. For example, the absolute value of both 5 and -5 is 5. This is denoted using vertical bars, like this: \( |x| \). Understanding absolute values is crucial when dealing with inequalities involving absolute values because it ensures we can correctly express the distance from zero.

compound inequality

A compound inequality involves two separate inequalities that are combined into one statement by the words 'and' or 'or'. For example, the compound inequality \(a < x < b\) means that x is between a and b, including all the values in that range. On the other hand, the inequality \(x < a \text{ or } x > b\) means x is either less than a or greater than b, excluding the values in the range. When working with absolute values, we often split the original inequality into two separate inequalities (compound inequality) to simplify the problem into manageable parts.

steps to isolate absolute value

Isolating the absolute value expression is the first and perhaps most critical step when solving absolute value inequalities. To isolate the absolute value, follow these steps:

• Identify the absolute value expression in the inequality. For example, in \(|0.5x - 3.5| + 0.2 \geq 0.6\), the absolute value expression is \(|0.5x - 3.5|\).
• Remove any constants added to or subtracted from the absolute value term by performing inverse operations on both sides of the inequality. For instance, subtract 0.2 from both sides to get \( |0.5x - 3.5| \geq 0.4\).
• Once the absolute value is isolated, the inequality is simplified, making it easier to break down into separate inequalities for further solving.

solving inequalities

Solving inequalities is about finding the values of the variable that make the inequality true. When working with absolute value inequalities, here are the steps:

• After isolating the absolute value, split the inequality into two cases - one for the positive and one for the negative. For instance, \(|0.5x - 3.5| \geq 0.4\) becomes two separate inequalities: \(0.5x - 3.5 \geq 0.4\) and \(0.5x - 3.5 \leq -0.4\).
• Solve each inequality separately. For the first one, \(0.5x \geq 3.9\), divide both sides by 0.5 to find \(x \geq 7.8\). Similarly, solve the second inequality \(0.5x \leq 3.1\) to find \(x \leq 6.2\).
• Combine the solutions using the appropriate conjunction. In this case, since we are dealing with a 'greater than or equal to' inequality, the final solution is \(x \leq 6.2 \text{ or } x \geq 7.8\).

It's important to practice these steps to become comfortable with solving absolute value inequalities efficiently and accurately.