Problem 20 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2 x+4 \leq 4 $$

Step-by-Step Solution

Answer

The solution is \(x \geq 0\), or in interval notation \([0, \infty)\).

Step 1: Simplify the Inequality

First, simplify the given inequality \(-2x + 4 \leq 4\) by subtracting 4 from both sides. This results in the inequality \(-2x \leq 0\).

Step 2: Solve for x

To solve for \(x\), divide both sides of the inequality \(-2x \leq 0\) by \(-2\). Remember, dividing by a negative number reverses the inequality sign. Therefore, \(x \geq 0\).

Step 3: Express the Solution in Interval Notation

With \(x \geq 0\), this solution means \(x\) ranges from 0 to positive infinity. In interval notation, this is expressed as \([0, \infty)\).

Step 4: Graph the Solution Set

On a number line, represent \([0, \infty)\) by drawing a solid dot at 0 and shading the line to the right, indicating that all numbers greater than or equal to 0 are included in the solution set.

Key Concepts

Solution Set

In the realm of inequalities, the term **solution set** refers to the collection of values that satisfy the given inequality. It defines all possible values that a variable can assume to make the inequality true. In our specific example, we reached the conclusion that the solution to \(-2x + 4 \leq 4\) is that \(x \geq 0\).This means any value of \(x\) starting from 0 onwards satisfies the inequality. It's crucial to understand this set thoroughly, as it determines which numbers make the initial condition valid. In simple terms, once you solve an inequality, the solution set tells us the territory on the number line where true solutions live.

Interval Notation

**Interval notation** is a mathematical representation used to define the set of solutions in a precise manner. It is particularly helpful for visually conveying which numbers are included in the solution set of an inequality.
For our inequality solution \(x \geq 0\), we express the solution set in interval notation as \([0, \infty)\).

• The square bracket \([\) at 0 signifies that 0 is included in the set.
• The parenthesis \()\) next to infinity indicates that infinity is an idea rather than a number, thus it’s never included.

Interval notation simplifies the expression of solutions by eliminating the need for complex descriptions while maintaining clarity and precision.

Graphing Inequalities

When dealing with inequalities, visualizing the solution on a number line is a powerful tool. This is what we refer to as **graphing inequalities**. This process captures the essence of the solution set, making it easier to comprehend which numbers make the inequality true.
In our solved inequality \(x \geq 0\), we illustrate the solution on a number line by placing a solid dot at 0. The solid dot indicates that 0 itself is part of the solution. Then, we shade the number line to the right of 0 to symbolize all numbers greater than or equal to 0.

• The solid dot denotes inclusion of the endpoint in the solution.
• Shading conveys all possible solutions moving to positive infinity.

Graphing helps students and mathematicians alike intuitively understand the concept of inequality solutions, serving as a bridge between symbolic and visual forms of mathematical expressions.