Question

Solve the inequality and graph its solution. \(x+6<8 \quad\)

Short answer

The solutions of the inequality \(x + 6 < 8 \) are all real numbers less than 2, graphically represented on a number line with all points to the left of 2 shaded, and the point at 2 being open.

Step-by-step solution

Subtract 6 from both sides

This step involves subtracting 6 on both sides of the inequality. This can be mathematically represented as: \(x + 6 - 6 < 8 - 6 \) which simplifies to \(x < 2 \).

Graph the result

After deducing that \(x < 2 \), the solution must be represented graphically. This involves drawing a number line, marking the number 2 on it, and shading all points to the left of 2, to indicate that those are the valid solutions for \(x \). The point at 2 should be open (not filled), as the inequality does not include 2 itself.

Key concepts

Inequality Graphing

Inequality graphing is a visual way to represent the solutions of an inequality. Just as with equations, inequalities have solutions which form a range of values rather than a single number. To graph an inequality like x + 6 < 8, we begin by solving it algebraically, as outlined in the textbook solution, resulting in the simplified form x < 2.

To graph this on a number line, follow these steps:

• Draw a horizontal line, which is your number line.
• Mark integers on the line, ensuring the critical value on the inequality, which is 2, is displayed.
• Place an open circle (not filled in) at 2 to show that this value is not included in the solution set.
• Shade the number line to the left of the open circle, indicating all numbers less than 2 are included in the solution set.

The open circle is key to indicating that the exact value 2 is not part of the solution, contrasting with a closed circle that would be used if the inequality was ≤ (less than or equal to). The shading direction (left for less than; right for greater than) provides an immediate visual cue about the solution set.

Algebraic Manipulation

Algebraic manipulation is the process of arranging and simplifying algebraic expressions in order to solve for a specific variable. In the context of solving inequalities, these skills are at the heart of finding the solution to an inequality. It involves following the same rules you'd use for equations while being cautious about the special rule—which is if you multiply or divide by a negative number, you must flip the direction of the inequality sign.

Looking at our exercise, x + 6 < 8, the first step, as mentioned in the solution, is subtracting 6 from both sides to isolate x. This is represented as:

• Original inequality: x + 6 < 8
• Subtract 6: (x + 6) - 6 < 8 - 6
• Simplified inequality: x < 2

Always perform the same operation on both sides of the inequality to maintain the balance. This principle ensures that the inequality remains true and valid throughout the manipulation process.

Number Line Representation

The number line representation is a critical tool in visually understanding the range of values for which an inequality is valid. It serves as a simple yet powerful way to communicate the solution of an inequality.

After solving x < 2 through algebraic manipulation, representing this on a number line solidifies the concept. It effectively communicates the idea that any number to the left of 2 on the number line is a solution to the inequality. Here's how you can do it:

• Use a straight edge to draw a number line on paper.
• Inscribe short vertical marks to represent numbers, ensuring that you include 0 (the origin), the value of 2, and numbers on either side of 2.
• Place an open circle over 2 to signal that 2 is not a solution.
• Shade to the left of the open circle to demonstrate that all numbers less than 2 are solutions to the inequality.

Thus, the number line should extend in both directions, but the shading will only cover the part that represents the solution set, which in this case are all the numbers lower than 2.