Problem 25 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$4 x<12$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 3\). In the graphical representation, a hollow circle is placed at 3, and an arrow pointing to the left represents all numbers less than 3.

Step 1: Solve the inequality

Let's start by isolating \(x\) on one side of the inequality. We do this by dividing both sides of the inequality by 4. Doing so gives us the inequality \(x < 3\).

Step 2: Graph the inequality

To graph this inequality on a number line, we start by placing a hollow circle at 3 on the number line. The hollow circle indicates that 3 is not part of the solution to the inequality \(x < 3\). Then, draw an arrow pointing to the left from 3 to indicate that any number less than 3 is a solution to the inequality.

Key Concepts

Solving Inequalities

Solving inequalities is not so different from solving equations, but there are some key points to keep in mind. Just like an equation, you want to isolate the variable, in this case, "x," on one side of the inequality.
To solve the inequality \(4x < 12\), the goal is to find out what values of "x" make the inequality true.

• Identify the operation with the variable. Here, 4 is multiplied by x.
• Use the inverse operation to isolate "x". In this case, divide both sides by 4 to get \(x < 3\).

It's important to note, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign. This is not the case here since we're dividing by a positive number, so the sign remains the same.

Graphing Inequalities

Graphing inequalities involves representing the solution on a number line. Once you solve the inequality, like \(x < 3\), you need to show which values "x" can take. Start by placing a circle on the number line at the boundary point, in this instance, 3.
Since the inequality is strictly less than 3, use an open circle to show that 3 itself is not included in the solution. Then, since you are interested in all values less than 3, draw a line with an arrow to the left of the number line.
This tells us that all numbers to the left of 3 satisfy the inequality \(x < 3\).

Number Line Representation

Number line representation is a pivotal concept in graphing inequalities. It provides a visual way of presenting which numbers are part of the solution.

Using Open and Closed Circles

• An open circle represents numbers not included in the solution set. This is used for inequalities with "<" or ">".
• A closed circle signifies that the boundary number is included, used with "\(\leq\)" or "\(\geq\)".

Remember to accompany the open circle with a direction on the number line, using arrows to effectively communicate the range of solutions.

Algebra Concepts

Understanding algebra concepts is crucial when solving inequalities or any mathematical equations. These fundamental concepts allow you to manipulate inequalities into a form that’s easy to graph or interpret.

Inverse Operations

Using inverse operations is a key algebraic concept. To move a term from one side of an inequality to the other, perform the opposite operation:

• Addition becomes subtraction.
• Multiplication becomes division.

With inequalities, always pay attention to the direction of the inequality symbol. Multiplying or dividing by a negative number requires you to flip the sign, an important rule that differentiates inequalities from equations. Understanding these basics will provide a strong foundation in algebra and set the stage for more complex problem-solving.