Question

Solve the inequality. Then graph the solution set on the real number line. $\left|\frac{2x+1}{2}\right|<6$

Short answer

The solution to the inequality $\left|\frac{2x+1}{2}\right|<6$ is the interval (-6.5, 5.5).

Step-by-step solution

Rewrite the inequality in standard form

Express the inequality as -6 < \frac{2x+1}{2} < 6. This is done according to the property of the absolute value mentioned above.

Solve the inequalities

Solve the inequalities separately. For the first one multiply each side by 2 and then subtract 1 from each side to get -13 < 2x or -6.5 < x. For the second one, again multiply each side by 2 and then subtract 1 to get 2x < 11 or x < 5.5.

Interpret the solution

Our solution to the original problem are values of x that satisfy both inequalities. The two solution intervals we have are (-6.5, ∞) and (-∞, 5.5). The overlap of these two intervals (-6.5, 5.5) is our solution to the original inequality.

Graph the solution set

Plot the solution set on a number line. Mark the intervals (-6.5, 5.5) on the number line. Since the inequality does not include the endpoints, hollow circles are used to denote that the endpoints are not part of the solution set.

Key concepts

Absolute Value Properties

Understanding absolute value properties is crucial when solving inequalities involving absolute values. The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. For example, both $4$ and $-4$ have an absolute value of $4$, represented as $|4|=|-4|=4$.

The key property used in solving absolute value inequalities like $\left|\frac{2x+1}{2}\right|<6$ is the definition itself: $|a|<b$ implies that $-b<a<b$, meaning that the value inside the absolute value sign must be greater than $-b$ and less than $b$. Conversely, for $|a|>b$, the value inside the absolute value must be either less than $-b$ or greater than $b$, creating two separate ranges to consider. These properties are the cornerstone of translating an absolute value inequality into a compound linear inequality.

Inequality Graphing

Graphical interpretation is a powerful tool for visualizing and solving inequalities. When graphing a linear inequality, we essentially plot the region that satisfies the inequality on the Cartesian plane.

However, since absolute value inequalities involve distances from zero, they are best represented on a number line. From the exercise $\left|\frac{2x+1}{2}\right|<6$, the conversion into two linear inequalities results in the double inequality $-6<\frac{2x+1}{2}<6$. By graphing these inequalities on a number line, we create a visual representation that helps us better understand the solution set. When graphing, it's essential to use open or closed dots to indicate whether endpoints are included in the solution (closed dots) or not (open dots). This attention to detail ensures that solutions are accurately represented.

Number Line Representation

The number line is a straightforward yet powerful mathematical tool for representing real numbers and their relationships, especially when it comes to inequalities.

For absolute value inequalities, after we have turned the complex inequality into simpler ones, as done with our example, we then use the number line to demonstrate the solution set. The key is to mark the critical points, which in our exercise are $-6.5$ and $5.5$, accurately on the line. We indicate that these points do not belong to the solution set by placing hollow circles at these locations, as the inequality is strict (\

Linear Inequality Solutions

Solving linear inequalities is similar to solving linear equations with the added consideration of the inequality sign. For the given exercise $\left|\frac{2x+1}{2}\right|<6$, the absolute value inequality is first portrayed as a compound linear inequality. Separating them gives us two distinct linear inequalities to solve: $-13<2x$ which simplifies to $x>-6.5$, and $2x<11$ which simplifies to $x<5.5$.

The real solution is the intersection of these two sets, meaning that $x$ must satisfy both conditions simultaneously. This is found by identifying the common values of $x$ that are true for both inequalities, resulting in our final solution, $-6.5<x<5.5$. Insight into linear solutions not only aids in resolving individual inequalities but also in understanding the comprehensive solution when addressing absolute value inequalities.