Problem 79 - 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) \(1-2 x=0\) (b) \(1-2 x \leq 0\) (c) \(1-2 x \geq 0\)

Step-by-Step Solution

Answer

a) \(x = \frac{1}{2}\); b) \([\frac{1}{2}, \infty)\); c) \((-\infty, \frac{1}{2}]\).

Step 1: Solve the Equation

Let's solve the equation \(1 - 2x = 0\). To find the value of \(x\), isolate \(x\) by moving \(1\) to the right side:\(1 = 2x\).Now, divide both sides by \(2\) to solve for \(x\):\(x = \frac{1}{2}\).

Step 2: Solve the Inequality (b)

Solve the inequality \(1 - 2x \leq 0\) by first isolating \(-2x\) on one side:\(-2x \leq -1\)Now, divide every term by \(-2\). Remember that dividing by a negative number reverses the inequality sign:\(x \geq \frac{1}{2}\).So, the solution in interval notation is \([\frac{1}{2}, \infty)\).

Step 3: Solve the Inequality (c)

Solve the inequality \(1 - 2x \geq 0\) again by isolating \(-2x\):\(-2x \geq -1\)Divide each term by \(-2\) and reverse the inequality:\(x \leq \frac{1}{2}\).Thus, the solution in interval notation is \((-\infty, \frac{1}{2}]\).

Key Concepts

Equations

In precalculus, equations play a crucial role in finding unknown variables. An equation states that two expressions are equal and involves an equality sign, like in the equation \(1 - 2x = 0\).

To solve an equation, the goal is to isolate the variable on one side to find its value. For linear equations, like \(1 - 2x = 0\), follow these steps:

• Move constant terms to one side of the equation, here we would move \(1\) by subtracting it from both sides, resulting in \(2x = 1\).


• Divide each side to isolate \(x\). Since we have \(2x = 1\), divide by \(2\) to get \(x = \frac{1}{2}\).

Solving equations like these helps to lay a strong foundation for understanding more complex algebraic expressions.

Inequalities

Inequalities are similar to equations but instead of an equality sign, they use symbols like \( \leq, \geq, <, \) and \(>\). These symbols indicate the relationship between two expressions—whether one is less than or greater than another. For example:

• \(1 - 2x \leq 0\) implies \(-2x\) is less than or equal to \(-1\).
• \(1 - 2x \geq 0\) suggests \(-2x\) is greater than or equal to \(-1\).

To solve inequalities, follow similar steps as solving equations but remember to flip the inequality sign when you multiply or divide by a negative number:

• For \(1 - 2x \leq 0\), isolate \(-2x\) to find \(-2x \leq -1\), and divide by \(-2\), which flips the sign to \(x \geq \frac{1}{2}\).
• For \(1 - 2x \geq 0\), \(x \leq \frac{1}{2}\) after isolating and solving similarly.
  

Understanding these rules allows you to determine a range of possible solutions that satisfy the inequality.

Interval Notation

Interval notation is a convenient method for expressing a range of values that satisfy certain conditions, particularly in inequalities. It uses brackets to describe intervals which are part of a set of possible values. Here's how it works:

• \([a, b]\) indicates all numbers between \(a\) and \(b\), inclusive of both \(a\) and \(b\).


• \((a, b)\) shows numbers between \(a\) and \(b\) but excludes \(a\) and \(b\).


• \([a, b)\) includes \(a\) but excludes \(b\), while \((a, b]\) does the opposite.

For inequalities:

• The solution \(x \geq \frac{1}{2}\) is expressed in interval notation as \([\frac{1}{2}, \infty)\), indicating all values starting from \(\frac{1}{2}\) going to infinity.
• Similarly, \(x \leq \frac{1}{2}\) states \(( -\infty, \frac{1}{2}]\), showing every value from negative infinity up to and including \(\frac{1}{2}\).

Interval notation simplifies expressing solution sets as compact intervals, making it easier to communicate the range of possible solutions.