Problem 80 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ -1=1-|0.1 x+8| $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -60\) and \(x = -100\).

Step 1: Isolate the Absolute Value Expression

The equation is \(-1 = 1 - |0.1x + 8|\). First, we need to isolate the absolute value expression. Subtract 1 from both sides to get:\[-1 - 1 = -|0.1x + 8|\]This simplifies to:\[-2 = -|0.1x + 8|\].

Step 2: Remove the Negative Sign

To simplify, we multiply both sides of the equation by -1:\[2 = |0.1x + 8|\].Now the absolute value is isolated and positive.

Step 3: Solve the Absolute Value Equation

The absolute value equation \(2 = |0.1x + 8|\) splits into two separate linear equations:1. \(0.1x + 8 = 2\)2. \(0.1x + 8 = -2\).Next, we'll solve each equation separately.

Step 4: Solve the First Equation

Solve the equation \(0.1x + 8 = 2\):1. Subtract 8 from both sides:\[0.1x = -6\]2. Divide both sides by 0.1:\[x = \frac{-6}{0.1} = -60\].

Step 5: Solve the Second Equation

Now solve the equation \(0.1x + 8 = -2\):1. Subtract 8 from both sides:\[0.1x = -10\]2. Divide both sides by 0.1:\[x = \frac{-10}{0.1} = -100\].

Step 6: Write the Solutions

The solutions to the equation are \(x = -60\) and \(x = -100\).

Key Concepts

Linear Equations

A linear equation is any equation that can be written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. It involves a linear expression, meaning it does not contain variables raised to a power other than one. Linear equations generally describe a straight line when graphed.

In solving linear equations, the goal is to isolate the variable, usually by moving all terms containing the variable to one side of the equation and constant terms to the other. This often involves using addition, subtraction, multiplication, or division.

• For instance, in the steps provided: by subtracting 8 from both sides, we simplify \(0.1x + 8 = 2\) to \(0.1x = -6\).
• The next step is to divide by 0.1 to isolate \(x\), resulting in \(x = -60\). The same process is followed for the second equation derived from the absolute value condition.

Understanding these operations is fundamental in working with a broader range of mathematical problems, including those involving absolute values.

Inequalities

Inequalities are statements about the relative size or order of two numbers or expressions. They are similar to equations, but instead of equality, symbols like \(<\), \(>\), \(\leq\), or \(\geq\) are used. Solving these equations involves finding the set of values for a variable that satisfies the inequality condition.

When working with inequalities, you perform similar operations as with equations:

• If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, multiplying \(\-x > 3\) by \(-1\) will result in \(x < -3\).
• Inequalities can often be visually represented using a number line or graphically to understand the solution set better.

By practicing with inequalities, you develop a better understanding of conditions that involve ranges of possible values, a skill also handy in probability and optimization problems.

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. It captures information about the boundaries of the set; whether the bounds are included or not depends on the use of brackets or parentheses.

Here's a quick guide to understanding interval notation:

• "(" or ")": These symbols indicate that the boundary is not included, known as an open interval.
• "[" or "]": These symbols indicate that the boundary is included, known as a closed interval.

For example, the notation \((2, 5]\) means values greater than 2 and up to and including 5. This is an efficient way to express solutions to inequalities and describe the domain of functions.

For absolute value equations leading to inequalities, once solutions are found, interval notation can clearly communicate the solution set. It provides a more formal method of expressing ranges of numbers, essential for calculus and analysis.