Problem 79 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ -7=2-|0.3 x-3| $$

Step-by-Step Solution

Answer

The solutions are x = 40 and x = -20.

Step 1: Isolate the Absolute Value

Start by isolating the absolute value on one side of the equation. In our equation, we have:\[-7 = 2 - |0.3x - 3|\]Subtract 2 from both sides to isolate the absolute value:\[-7 - 2 = - |0.3x - 3|\]This simplifies to:\[-9 = - |0.3x - 3|\]Multiply both sides by -1 to remove the negative sign from the absolute value:\[9 = |0.3x - 3|\]

Step 2: Rewrite the Absolute Value Equation

The equation \( |0.3x - 3| = 9 \) means that there are two possible equations to consider:1. \( 0.3x - 3 = 9 \)2. \( 0.3x - 3 = -9 \)

Step 3: Solve the First Equation

Take the first equation:\[0.3x - 3 = 9\]Add 3 to both sides:\[0.3x = 12\]Divide by 0.3:\[x = \frac{12}{0.3} = 40\]So, one solution is \( x = 40 \).

Step 4: Solve the Second Equation

Take the second equation:\[0.3x - 3 = -9\]Add 3 to both sides:\[0.3x = -6\]Divide by 0.3:\[x = \frac{-6}{0.3} = -20\]So, the second solution is \( x = -20 \).

Step 5: Verify the Solutions

Substitute \( x = 40 \) and \( x = -20 \) back into the original equation to verify. For simplicity, both calculations result in a check if the absolute value equals 9:- For \( x = 40: \)\( |0.3(40) - 3| = |12 - 3| = 9 \)- For \( x = -20: \)\( |0.3(-20) - 3| = |-6 - 3| = 9 \)Both solutions satisfy the original equation.

Key Concepts

Solving Absolute Value Equations

Solving absolute value equations might seem tricky at first, but with a few clear steps, you'll master it in no time.
At the heart of every absolute value equation is the understanding that absolute value represents the distance of a number from zero on the number line. This means it is always non-negative. For example, in our exercise, the absolute value part is \(|0.3x - 3|\).
To solve the equation \(|0.3x - 3| = 9\), you should consider that the expression inside can equal either 9 or -9. This is because both numbers, when measured for their distance from zero, give the same absolute value.

• You first break it down into two separate equations: 1) \(0.3x - 3 = 9\) 2) \(0.3x - 3 = -9\)
• Now solve each of these equations separately to find the values of \(x\).

This process shows us that every absolute value equation potentially has two solutions.

Graphing Inequalities

When it comes to graphing inequalities, you begin by thinking about what the inequality is communicating. For example, with the inequality found from solving \(|0.3x - 3| \leq 9\), you want to know which values of \(x\) satisfy this condition.
To graph inequalities, start by transforming the inequality into two separate statements, similar to solving absolute value equations. Then consider:

• The solution from the inequality \(0.3x - 3 \leq 9\)
• The solution from the inequality \(0.3x - 3 \geq -9\)

After solving, you plot these intervals on a number line. For simplicity, use shading or a bold line to indicate all solutions that satisfy the inequality. Remember, when graphing:

• A closed circle is used if the interval includes the endpoint.
• An open circle is used if the endpoint is not included.

This method clearly indicates the range of solutions visually.

Interval Notation

Interval notation is a concise way to describe a set of numbers, especially useful for expressing solutions to inequalities. Once you've solved an inequality and found the range of solutions, you'll want to express this using interval notation.
Suppose the solutions are \(x \geq 3\) and \(x \leq 12\). In interval notation, this is written as \([3, 12]\). Note that:

• Round brackets \((\) are used if the endpoint is not included.
• Square brackets \([\) are used if the endpoint is included.

Comprehensive interval notation can easily convey a set of solutions without the need for bulky descriptive language, making it a favorite among mathematicians for its efficiency.

Isolating Absolute Values

The first step in solving any absolute value equation is to isolate the absolute value expression. The goal here is to get the absolute value by itself on one side of the equation so you can then proceed with breaking it into two separate linear equations.
In this exercise, we started with \(-7 = 2 - |0.3x - 3|\).
To isolate, subtract 2 from both sides and then multiply by -1 to keep the absolute value positive, resulting in \(9 = |0.3x - 3|\).
Isolating absolute values is crucial because it sets the stage for applying the principle that the expression inside the bars could equal either the positive or negative value of the constant on the other side of the equation.
This consistent process forms the foundation of accurately solving absolute value equations.