Problem 87 - Algebra for College Students

Question

Solve each equation or inequality. $$ |0.1 x-2.5|+0.3 \geq 0.8 $$

Step-by-Step Solution

Answer

The solution to the inequality is \( 20 \ x \ 30 \).

Step 1: Simplify the Inequality

First, isolate the absolute value term. Subtract 0.3 from both sides of the inequality:\[ |0.1x - 2.5| + 0.3 - 0.3 \ \rightarrow |0.1x - 2.5| \ \rightarrow 0.8 - 0.3 = 0.5 \]This simplifies to:\[ |0.1x - 2.5| \ \rightarrow 0.5 \]

Step 2: Formulate Two Separate Inequalities

To solve the absolute value inequality \( |0.1x - 2.5| \ \rightarrow 0.5 \), consider the two cases:\[ 0.1x - 2.5 \ \rightarrow 0.5 \ \text{and} \ 0.1x - 2.5 \ \rightarrow -0.5 \]

Step 3: Solve the First Inequality

Solve the inequality \(0.1x - 2.5 \ \rightarrow 0.5\):Add 2.5 to both sides:\[ 0.1x - 2.5 + 2.5 \ \rightarrow 0.5 + 2.5 \]This simplifies to:\[ 0.1x \ \rightarrow 3 \]Divide both sides by 0.1:\[ x \ \rightarrow \frac{3}{0.1} = 30 \]

Step 4: Solve the Second Inequality

Solve the inequality \(0.1x - 2.5 \ \rightarrow -0.5\):Add 2.5 to both sides:\[ 0.1x - 2.5 + 2.5 \ \rightarrow -0.5 + 2.5 \]This simplifies to:\[ 0.1x \ \rightarrow 2 \]Divide both sides by 0.1:\[ x \ \rightarrow \frac{2}{0.1} = 20 \]

Step 5: Combine the Solutions

Combine the solutions from steps 3 and 4 to form the interval:\[ 20 \ x \ 30 \]

Key Concepts

solving absolute value equations

When dealing with absolute value equations, it is crucial to understand what absolute value represents. The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, both 3 and -3 have an absolute value of 3.

To solve an absolute value equation like \(|Ax + B| = C\), you break it down into two separate equations:

• \(Ax + B = C\)
• \(Ax + B = -C\)

This translates the absolute value condition into a form that can be solved using standard algebraic methods. Each of these equations can be solved separately by isolating the variable through addition, subtraction, multiplication, or division.

After solving both equations, you can combine the solutions to find the interval or specific values that satisfy the original equation.

inequalities

Inequalities describe a relationship where one side is not simply equal to the other, but either greater, less than, or some combinations thereof. Solving inequalities is similar to solving equations, but with particular care for the direction of the inequality:

• When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

For instance, if \[Ax + B \rightarrow C\], then two separate inequalities can be formed for the absolute value conditions. These inequalities give ranges of values that satisfy the original condition. The results can then be expressed in interval notation.

interval notation

Interval notation is a concise way to express a range of values. Instead of listing all solutions individually, interval notation uses brackets and parentheses to describe sets:

• Use square brackets [ ] to indicate that an endpoint is included (closed interval).
• Use parentheses ( ) to indicate that an endpoint is not included (open interval).

For example, from the given solution 20 < x < 30, in interval notation, this means \( (20, 30) \).

Combining solutions from absolute value inequalities, such as \(x \rightarrow 30\) and \(x \rightarrow 20\), means the values of x should be within the range, forming an interval that encompasses both results.