Question

Solve the inequality. Then graph the solution set on the real number line. \(\left|\frac{x}{2}\right|>3\)

Short answer

The solution for the inequality is \(x > 6\) or \(x < -6\) when plotted on a number line.

Step-by-step solution

Understand Absolute Value

Understand that an absolute value |a| represents a distance, and is always positive, except for zero. So, in the absolute value inequality, |x/2| > 3, x/2 can either be greater than 3 or less than -3.

Create Two separate Inequalities

From this understanding, create two separate inequalities: one for when x/2 is greater than 3, and one for when x/2 is less than -3. So, the inequalities will be: x/2 > 3 and x/2 < -3.

Solve each Inequality

Solve each inequality separately. This involves multiplying each side by 2. The first inequality, x/2 > 3, becomes x > 6 when we multiply each side by 2. Similarly, the second inequality, x/2 < -3, becomes x < -6.

Graph the solution on the Real number line

The solution to the inequality is the union of all x that satisfy either x > 6 or x < -6. To graph this, simply draw a number line and shade in all points to the right of 6 and to the left of -6. Make sure to use an open circle on 6 and -6, as these are not in the solution set.

Key concepts

Inequality Solving

To solve an absolute value inequality like \(\left|\frac{x}{2}\right| > 3\), you need to understand the concept of absolute value. Absolute value represents distance from zero, which is always a non-negative number. In this inequality, \(\left|\frac{x}{2}\right| > 3\) means that the distance of \(\frac{x}{2}\) from zero is greater than 3. This creates two separate conditions to consider:

• \(\frac{x}{2} > 3\)
• \(\frac{x}{2} < -3\)

Now, solve each inequality separately:

• For \(\frac{x}{2} > 3\), multiply both sides by 2 to get \(x > 6\).
• For \(\frac{x}{2} < -3\), multiply both sides by 2 to get \(x < -6\).

The solution to the original inequality \(\left|\frac{x}{2}\right| > 3\) is the union of \(x > 6\) or \(x < -6\). This logical approach ensures you capture all possible values of \(x\) that satisfy the inequality. Then, you can represent this solution visually on a number line.

Graphing on Number Line

Graphing the solutions of inequalities like \(x > 6\) or \(x < -6\) on a number line helps visualize the solution set. Start by drawing a horizontal line, which will serve as your number line. Identify the critical points, which are 6 and -6 in this case. Place marks at these points. However, since the solution is strict (greater or lesser, not equal), use open circles on these points, indicating they aren't included in the solution set.Next, shade the region to the right of 6, and the region to the left of -6, to show \(x > 6\) and \(x < -6\) respectively. This visual representation offers an immediate understanding of which numbers \(x\) can take, emphasizing the concept of all numbers greater than 6 and all numbers less than -6 being part of the solution.

Distance and Absolute Value

Understanding the connection between distance and absolute value is crucial for solving absolute value inequalities. The absolute value of a number gives its distance from zero on a number line. Therefore, saying \(\left|\frac{x}{2}\right| > 3\) implies that the value \(\frac{x}{2}\) is more than 3 units away from zero. This means moving either more than 3 units to the right or more than 3 units to the left from zero.Think of it as saying that \(\frac{x}{2}\) is not only beyond 3 but also beyond -3 in terms of distance. When visualized, this reveals values either beyond positive or negative bounds.Understanding absolute value in terms of distance makes these inequalities intuitive since visualizing quantities in terms of how far away they are from zero can simplify comprehension and problem-solving.