Problem 12 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ x-3<-4 $$

Step-by-Step Solution

Answer

The solution is \(x < -1\) and in interval notation: \((-\infty, -1)\).

Step 1: Identify the Inequality

The original inequality given is \(x - 3 < -4\). The task is to solve for \(x\).

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the inequality. Add 3 to both sides of the inequality to achieve this:\[x - 3 + 3 < -4 + 3\]This simplifies to:\[x < -1\]

Step 3: Express the Solution in Interval Notation

The solution \(x < -1\) is the set of all numbers less than -1. In interval notation, this is expressed as:\((-\infty, -1)\).

Step 4: Graph the Solution

On a number line, we represent the solution \(x < -1\). To do this, draw an open circle at -1 (to show that -1 is not included) and shade the line to the left of -1 to indicate all numbers less than -1.

Key Concepts

Solution Set

When dealing with inequalities, the solution set is crucial. It represents all possible values of a variable that satisfy the given inequality. In the inequality \(x - 3 < -4\), the goal was to solve for \(x\). This means finding all \(x\) values that make the inequality true.

To achieve this, we performed basic algebraic manipulations to isolate \(x\). By adding 3 to both sides, we simplified the inequality to \(x < -1\).

This indicates that any number less than \(-1\) will satisfy the original inequality. Solution sets can be thought of as a collection or a range of numbers that work for the inequality. It’s important because it gives a clear understanding of what numbers are possible solutions.

Interval Notation

Interval notation is a way of writing down subsets of the real number line. It's compact and efficient, giving us a clear picture of which numbers are included in a solution set under inequalities.

In our example, the solution \(x < -1\) translates to all real numbers that are less than \(-1\). In interval notation, this becomes \(( -\infty, -1)\).

Here's how it works:

• The parenthesis \((\)) indicates that the number isn’t included in the set. For \(-1\), we use \(()\) because \(-1\) itself isn't a solution.
• \(-\infty\) is always used with a parenthesis because infinity is a concept, not a number we can actually reach.

Interval notation is a helpful tool because it allows you to quickly see the range of solutions and the bounds of your solution set.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solution set, which can make it easier to understand where the solutions lie relative to zero or any other reference points.

For the inequality \(x < -1\), graphing involves these steps:

• Draw a horizontal line, known as the number line.
• Place an open circle on \(-1\). The open circle indicates that the number \(-1\) itself is not included in the solution set.
• Shade the area to the left of \(-1\). This shading represents all the numbers that are less than \(-1\).

The graph effectively communicates which values are solutions and which are not, making it an invaluable tool for both simple and complex inequalities. In essence, it turns the abstract concept of inequalities into a concrete image.