Problem 23 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$x+4 \leq 6$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq 2\). This means \(x\) can be any real number that is less than or equal to \(2\). The graphical representation of the solution set includes all real numbers to the left of and including \(2\) on a number line.

Step 1: Isolate the variable

In order to isolate \(x\), subtract \(4\) from both sides of the inequality \(x+4 \leq 6\). This operation does not change the direction of the inequality. The resulting equation is \(x \leq 2\).

Step 2: Graph the solution

The inequality \(x \leq 2\) suggests that all values of \(x\) that are less than or equal to \(2\) satisfy the inequality. Plot a number line from \(-5\) to \(5\) (for example), mark \(2\) on the number line, and shade all the region on the number line to the left of \(2\) including the point \(2\) itself. Put a closed circle at \(2\) to indicate that \(2\) is part of the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities is a handy way to visually represent solutions on a number line. It helps us see all the values that satisfy an inequality at a glance. When graphing the inequality \(x + 4 \leq 6\), we first simplify it to \(x \leq 2\) through algebraic methods. On a number line, this inequality includes all numbers less than or equal to 2.

To graph an inequality like \(x \leq 2\), follow these steps:

• Draw a horizontal line, which represents a range of possible values for \(x\).
• Mark the point \(2\) on this line. Since our inequality is \(\leq\), we use a closed circle around \(2\), indicating that 2 is a part of the solution set.
• Shade all the numbers to the left of the \(2\) on the line, showing that every number less than or equal to 2 meets the inequality's condition.

By following these steps, you'll create a clear and accurate graphical representation of the inequality's solutions.

Algebraic Solutions

Solving inequalities like \(x + 4 \leq 6\) algebraically is very similar to solving equations. The primary goal is to isolate the variable on one side to find a range of its possible values. Here, we'll break down how to reach the algebraic solution.

Start with the inequality:

• \(x + 4 \leq 6\)
• Subtract \(4\) from both sides of the inequality to isolate \(x\):
• \((x + 4) - 4 \leq 6 - 4\)
• This simplifies the inequality to \(x \leq 2\).

By performing this operation, the inequality sign remains unchanged because subtracting the same number from both sides doesn't affect the inequality's balance. The entire process ensures a clear algebraic solution - \(x\) can be any number equal or less than 2.

Number Line Representation

Using a number line to represent solutions is integral in understanding inequalities. It communicates which values satisfy the inequality very clearly.

For the inequality \(x \leq 2\), the number line helps illustrate the range of solutions:

• The number line includes integers and any other real numbers that lie along it.
• Identify the point \(2\) on the number line and place a closed circle. This circle signifies that \(2\) is included in the set of solutions.
• Shade or lightly draw a line extending left from \(2\) to indicate that all numbers less than \(2\) are also solutions.

This visual aid is particularly helpful for quickly checking your solution or when comparing multiple inequalities. It provides a straightforward, easily understood method for displaying all the potential solutions in one clear image.