Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(4-3 x=0\) (b) \(4-3 x \leq 0\) (c) \(4-3 x \geq 0\)

Step-by-Step Solution

Answer

(a) \(x = \frac{4}{3}\); (b) \([\frac{4}{3}, \infty)\); (c) \(( -\infty, \frac{4}{3}]\)

Step 1: Solving the Equation

We start with the equation given: \(4 - 3x = 0\). To solve for \(x\), we need to isolate \(x\) on one side of the equation. So, we'll move the constant to the other side: \(-3x = -4\). Then, divide both sides by \(-3\) to solve for \(x\): \(x = \frac{-4}{-3} = \frac{4}{3}\). Therefore, the solution to the equation is \(x = \frac{4}{3}\).

Step 2: Finding the Interval for the Inequality \(4-3x \leq 0\)

Start with the inequality: \(4 - 3x \leq 0\). We solve the inequality similarly to the equation. First, move the constant to the right: \(-3x \leq -4\). Now, divide by \(-3\) and remember to reverse the inequality sign when dividing by a negative number: \(x \geq \frac{4}{3}\). In interval notation, this solution is expressed as \([\frac{4}{3}, \infty)\). Therefore, \(x\) is greater than or equal to \(\frac{4}{3}\).

Step 3: Finding the Interval for the Inequality \(4-3x \geq 0\)

Consider the inequality: \(4 - 3x \geq 0\). Solving this inequality, move \(4\) to the right side to get \(-3x \geq -4\). Divide both sides by \(-3\), reversing the inequality sign: \(x \leq \frac{4}{3}\). The interval in which this inequality holds is \(( -\infty, \frac{4}{3}]\). Therefore, \(x\) is less than or equal to \(\frac{4}{3}\).

Key Concepts

Linear Equations

Linear equations are one of the simplest forms of equations and serve as a fundamental building block in algebra. A linear equation is an equation in which the highest power of the variable is one. Consider the general form of a linear equation: \[ ax + b = 0 \]

• Where \(a\) and \(b\) are constants, and \(x\) is the variable.
• The solution to a linear equation is the value of \(x\) that makes the equation true.

In our problem, we started with the equation:\[ 4 - 3x = 0 \]To solve this, we aim to isolate \(x\). By subtracting 4 from both sides, we get:\[ -3x = -4 \]Finally, divide both sides by \(-3\) to obtain:\[ x = \frac{4}{3} \]This solution means that when \(x\) is \(\frac{4}{3}\), the equation is satisfied.

Inequalities

Inequalities expand on the concept of equations by comparing expressions instead of stating they are equal. There are several inequality symbols we frequently encounter:

• \(\leq\) (less than or equal to)
• \(\geq\) (greater than or equal to)
• \(<\) (less than)
• \(>\) (greater than)

Solving inequalities often involves similar steps to solving equations, but with an important additional rule: when we multiply or divide both sides by a negative number, we must reverse the direction of the inequality.Let's look at the inequality \(4 - 3x \leq 0\) from our exercise:

• First, move the constant to the right side: \(-3x \leq -4\).
• Next, divide by \(-3\), reversing the inequality: \(x \geq \frac{4}{3}\).

The solution \(x \geq \frac{4}{3}\) indicates that \(x\) can be \(\frac{4}{3}\) or any number greater.

Interval Notation

Interval notation is a way to describe a range of values that satisfy an inequality. It uses brackets \([ \; ]\) and parentheses \(( \; )\) to indicate the inclusion or exclusion of endpoints. Here’s a quick guide:

• \([a, b]\) includes both endpoints \(a\) and \(b\).
• \((a, b)\) excludes both endpoints \(a\) and \(b\).
• A combination like \([a, b)\) includes \(a\) but not \(b\).
• \([-\infty, b)\) is an interval that extends infinitely leftward.
• \((a, \infty)\) extends infinitely rightward.

In the solutions to our inequalities:- The solution for \(4 - 3x \leq 0\) is \([\frac{4}{3}, \infty)\), indicating all \(x\) values from \(\frac{4}{3}\) to infinity.- Conversely, \(4 - 3x \geq 0\) transforms to \(( -\infty, \frac{4}{3}]\), capturing all values less than or equal to \(\frac{4}{3}\). This notation allows for a concise representation of the solutions to inequalities.