Problem 81 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. \((\text {Lesson } 6.1)\) $$ 2+x<4 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 2\). In the graph, an open circle is drawn at x = 2 and a ray is drawn to the left of 2.

Step 1: Isolate x in the inequality

To isolate x, subtract 2 from both sides of the inequality. This gives: 2 + x - 2 < 4 - 2, which simplifies to x < 2.

Step 2: Check the inequality

You can validate the solution by substituting a number less than 2 into the inequality for x and checking that it holds true. For example, if you substitute x = 1 into the inequality, you get 2 + 1 < 4, which is true, because 3 is less than 4.

Step 3: Graph the solution

On a number line, draw an open circle at x = 2 to represent that 2 is not included in the solution set (since x is less than, not equal to, 2). Next, draw a ray to the left of 2 to show all the numbers that are less than 2, which are solutions of the inequality.

Key Concepts

Solving Inequalities

When solving inequalities, the process is similar to solving regular equations. However, we must pay attention to the direction of the inequality sign. In the given exercise, we aim to isolate the variable \(x\) on one side of the inequality.

Here's a step-by-step method to solve simple inequalities:

• First, if necessary, use addition or subtraction to eliminate constants from the side of the inequality containing the variable.
• Second, use multiplication or division to get the coefficient of the variable to 1.

In our exercise, 2 is subtracted from both sides to isolate \(x\), resulting in the inequality \(x < 2\).

Note that this process maintains the direction of the inequality symbol unless you multiply or divide by a negative number. Doing so flips the inequality symbol, maintaining a true statement. Always check your solution by substituting back into the original inequality.

Graphing Inequalities

Visualizing the solutions of an inequality on a graph helps to understand it better. In the case of one-variable inequalities, we commonly represent them on a number line. The solution to the inequality \(x < 2\) is graphically represented with a number line graph.

Here’s how to graph it:

• Identify the number on the graph that makes the inequality an equality \(x = 2\).
• Draw a road sign, an open circle or hollow dot, at \(x = 2\), indicating this number is not included in the solution.

The open circle serves as a warning sign that we are excluding this point from our set of answers, reflecting the fact that \(x\) must be less than 2, not equal to 2.

Number Line Representation

The number line is a very helpful tool when working with inequalities. It allows you to visually represent the range of values for which an inequality holds true. For the inequality \(x < 2\), the number line representation involves drawing an open circle at number 2 and shading or drawing a ray extending to the left.

Let's elaborate on how this is done:

• The open circle at 2 signifies that the number 2 is not included, distinguishing it from solid circles used in \(\leq\) or \(\geq\) inequalities, where the boundary number is included.
• The ray extends to the left from 2, covering all values less than 2, effectively representing the solution set \(x < 2\).

A number line helps in understanding that the solution is not just a single number but rather a range, extending indefinitely in one direction. This visual representation is essential for quickly grasping which numbers are solutions to an inequality.