Problem 80 - A Graphical Approach to College Algebra

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: Solve the Equation 4 - 3x = 0

Start by isolating the variable x. To do this, subtract 4 from both sides to get:\[-3x = -4\]Next, divide each side by -3:\[x = \frac{-4}{-3} = \frac{4}{3}\]So, the solution for the equation is \(x = \frac{4}{3}\).

Step 2: Solve the Inequality 4 - 3x \leq 0

To solve \(4 - 3x \leq 0\), you start similarly by isolating x. Subtract 4 from both sides:\[-3x \leq -4\]Next, divide each side by -3. Remember, dividing by a negative number reverses the inequality:\[x \geq \frac{4}{3}\]So, the solution in interval notation is \([ \frac{4}{3}, \infty )\).

Step 3: Solve the Inequality 4 - 3x \geq 0

To solve \(4 - 3x \geq 0\), isolate x by subtracting 4 from both sides:\[-3x \geq -4\]Now, divide each side by -3, reversing the inequality:\[x \leq \frac{4}{3}\]The solution in interval notation is \(( -\infty, \frac{4}{3} ]\).

Key Concepts

Interval Notation

Interval notation is a way to represent a range of values where a variable lies. Instead of writing long inequalities to show these ranges, you can use a simpler format. Here's how it works:

• Use square brackets \( [ ] \) to indicate that an endpoint is included in the range (this is known as being a "closed" interval).
• Use parentheses \( ( ) \) to show that an endpoint is not included in the interval (an "open" interval).

For example, when the solution to an inequality is \(x \geq \frac{4}{3}\), it can be written in interval notation as \([ \frac{4}{3}, \infty)\). Here, the bracket indicates that \(\frac{4}{3}\) is part of the solution, whereas infinity always gets a parenthesis because it's a concept rather than a number.
Interval notation makes it clear and concise to express these solutions, making them easier to read and understand quickly.

Inequality Reversal

Reversing inequalities is crucial when solving them, especially when you multiply or divide both sides by a negative number. Here’s what you need to remember:

• In mathematics, multiplying or dividing both sides of an inequality by a negative number flips the inequality sign. For instance, \x \leq 3\ becomes \x \geq 3\ if multiplied by -1.
• Imagine an inequality as a balance scale. When you multiply or divide by negative values, the weights would tip the opposite way, thus the reversal.

This concept is especially important because if you forget to reverse the inequality, your solution will point in the wrong direction, making it incorrect.
A good example of this is seen in the inequality \(4-3x \leq 0\). After isolating the variable x, we divide by -3, reversing the \leq\ to \geq\.

Algebraic Isolation of Variables

Algebraic isolation of variables is a fundamental technique used in solving equations and inequalities. It involves getting the variable alone on one side of the equation or inequality so that you can easily determine its value.

Here's how you approach it:

• Identify the term containing the variable and move all other terms to the opposite side. Use addition or subtraction to achieve this.
• Next, if the variable is multiplied or divided by a coefficient, use division or multiplication to solve for the variable.

For example, in solving \(4 - 3x = 0\), the first step is to move the 4 to the other side by subtracting it. You then isolate \(x\) by dividing by -3.
This clear approach ensures that you maintain balance in the equation and can easily solve for any variable, making it a central strategy in algebra.