1. Technical Framework of the Absolute Value Inequality

Problem Parameters and Initial Variable Assignment

The problem under consideration requires the determination of the real number range for the variable ##x## such that the absolute value of the linear expression ##3x – 4## is greater than or equal to the constant integer ##11##. This specific mathematical structure is classified as a non-homogenous linear inequality involving an absolute value operator. In this technical context, we initialize the variable ##x## as a member of the set of real numbers ##\mathbb{R}##. The technical objective is to define the boundaries of the solution set that satisfies the condition ##|3x – 4| \ge 11## through rigorous algebraic manipulation.

In terms of initial conditions, the expression within the modulus bars, ##3x – 4##, is a first-order polynomial where the coefficient of the variable ##x## is ##3## and the constant term is ##-4##. The inequality sign ##\ge## indicates a non-strict, inclusive boundary condition, which suggests that the resulting solution set will involve closed intervals on the real number line. The magnitude of the threshold value is ##11##, representing the minimum distance from the origin required for the absolute displacement of the internal linear function. Establishing these parameters is critical before applying any branching logic to the expression.

Mathematically, the problem is represented as: ### |3x – 4| \ge 11 ### This inequality implies that the quantity ##3x – 4## must lie at a distance of at least ##11## units from the zero point on a one-dimensional coordinate system. Because the absolute value function always returns a non-negative result, the problem is well-posed for all real values of ##x##. We must address the inherent disjunction of the absolute value operator to isolate ##x##. This involves splitting the inequality into two distinct linear branches based on the fundamental definition of the modulus operator as a piecewise function.

Definition of the Piecewise Absolute Value Operator

The absolute value of an arbitrary real algebraic expression ##u##, denoted as ##|u|##, is defined by its piecewise behavior. Specifically, ##|u| = u## when ##u \ge 0##, and ##|u| = -u## when ##u < 0##. Applying this definition to our specific inequality ##|3x – 4| \ge 11##, we recognize that the solution space is partitioned into two regions of the real line. The first region occurs where the argument is non-negative, and the second occurs where it is negative. This piecewise mapping allows for the transformation of a single absolute value inequality into a compound statement of linear inequalities.

The logical structure of an absolute value inequality of the form ##|u| \ge a##, where ##a > 0##, is inherently a disjunction. This means the condition is satisfied if either ##u \ge a## or ##u \le -a## is true. In the context of our specific problem, this translates to the following logical disjunction: ### (3x – 4 \ge 11) \lor (3x – 4 \le -11) ### Each branch of this disjunction must be solved independently to find the component intervals of the complete solution set. This approach ensures that we capture all values of ##x## that result in an absolute magnitude of at least ##11##.

This methodology is grounded in the geometric interpretation of the absolute value as a metric for distance. The inequality states that the value ##3x – 4## must be located at or beyond the points ##11## and ##-11##. By converting the problem into two linear inequalities, we eliminate the non-linear nature of the absolute value operator, allowing for standard algebraic isolation of the variable. This step is the pivot point between the problem definition and the algorithmic computation of the results. It is essential to maintain the correct inequality signs throughout this conversion to ensure the validity of the final set-theoretic union.

2. Computational Execution of the Solution Branches

Analytical Resolution of the Positive Branch Linear System

The first linear branch to be analyzed corresponds to the scenario where the internal expression ##3x – 4## is greater than or equal to the positive threshold value of ##11##. The algebraic goal here is to isolate the variable ##x## using standard operations. We begin with the inequality: ### 3x – 4 \ge 11 ### To proceed, we apply the addition property of inequalities by adding ##4## to both sides of the expression. This operation maintains the direction of the inequality sign and isolates the term containing the variable on the left-hand side of the relation.

After adding ##4## to both sides, the expression simplifies as follows: ### 3x \ge 11 + 4 ### ### 3x \ge 15 ### In this intermediate stage, we have a simplified linear inequality where the product of ##3## and ##x## must be at least ##15##. The next logical step is to isolate ##x## by dividing both sides by the coefficient ##3##. Since ##3## is a positive real number, the division operation does not reverse the inequality sign, preserving the original logical direction of the branch. This is a crucial detail in maintaining the integrity of the solution.

The final step for the positive branch yields: ### x \ge \frac{15}{3} ### ### x \ge 5 ### This result indicates that for any real value of ##x## that is greater than or equal to ##5##, the original absolute value inequality will be satisfied. For example, if we substitute ##x = 6##, we obtain ##|3(6) – 4| = |14| = 14##, which is clearly greater than or equal to ##11##. This interval represents the right-hand portion of our dual-interval solution set. The technical validity of this branch is confirmed by its consistency with the initial positive domain assumption.

Analytical Resolution of the Secondary Inequality Branch

The second branch of the disjunction addresses the case where the expression ##3x – 4## is less than or equal to the negative threshold value of ##-11##. This represents the left-hand displacement on the coordinate line. We start with the following linear inequality: ### 3x – 4 \le -11 ### Similar to the previous branch, we utilize the addition property to move the constant term to the right-hand side. Adding ##4## to both sides of the inequality allows us to focus exclusively on the term containing the variable ##x##, facilitating the isolation process.

The simplification of this secondary branch proceeds as follows: ### 3x \le -11 + 4 ### ### 3x \le -7 ### At this juncture, we observe that ##3x## must be less than or equal to ##-7##. To solve for ##x##, we again divide both sides of the inequality by the positive constant ##3##. As noted previously, dividing by a positive number leaves the inequality sign ##\le## unchanged. This step is essential for correctly identifying the upper bound of the negative interval. Precision in sign management is paramount when dealing with negative constants in inequality systems to avoid common computational errors.

The final calculation for this branch is: ### x \le -\frac{7}{3} ### This result can also be expressed in decimal form as approximately ##x \le -2.333…##, though the fraction form is preferred for technical accuracy. This interval encompasses all real numbers from negative infinity up to and including ##-7/3##. To verify, choosing ##x = -3## gives ##|3(-3) – 4| = |-13| = 13##, which satisfies the condition ##13 \ge 11##. This secondary branch completes the algebraic derivation required to define the boundaries of the solution set within the real number system.

3. Set-Theoretic Synthesis and Geometric Mapping

Construction of the Unified Solution Interval Set

With the two independent branches solved, the next phase of the technical analysis involves synthesizing these results into a unified solution set. In set theory, the logical “or” connector used in the disjunction corresponds to the union operator ##\cup##. Therefore, the complete solution for ##x## is the union of the two sets derived from the linear branches. The final solution set in interval notation is expressed as: ### x \in (-\infty, -\frac{7}{3}] \cup [5, \infty) ### The use of square brackets indicates that the endpoints ##-7/3## and ##5## are included in the solution set, reflecting the non-strict nature of the original inequality.

This interval notation provides a concise and professional representation of the solution space. The first interval ##(-\infty, -7/3]## represents the left-hand unbounded region, while the second interval ##[5, \infty)## represents the right-hand unbounded region. Between these two intervals lies a gap from ##-7/3## to ##5## where the inequality is not satisfied. Specifically, for any ##x## in the open interval ##(-7/3, 5)##, the absolute value of ##3x – 4## will be strictly less than ##11##. This dual-interval structure is characteristic of “greater than or equal to” absolute value inequalities.

Alternatively, the solution can be presented as a set-builder notation for enhanced formal rigor. This is written as: ### \{ x \in \mathbb{R} \mid x \le -\frac{7}{3} \text{ or } x \ge 5 \} ### This notation explicitly defines the membership criteria for the set of real numbers that satisfy the problem. In professional mathematical documentation, providing the solution in multiple formats ensures clarity and prevents misinterpretation of the results. This synthesis phase transitions the analysis from algebraic manipulation to formal set-theoretic result presentation, which is essential for higher-level mathematical communication and further computational applications in calculus or engineering.

Number Line Visualization and Critical Value Verification

Visualizing the solution set on a real number line serves as a critical verification step to ensure geometric consistency. On the coordinate axis, the critical values are ##x = -7/3## and ##x = 5##. To represent the solution graphically, solid circles are placed at these two points to denote their inclusion in the set. Rays are then drawn extending outward: one starting at ##-7/3## and heading toward negative infinity, and another starting at ##5## and heading toward positive infinity. This visualization highlights the symmetry and displacement required by the absolute value operator.

Verification of these results involves testing values from each of the identified regions. We have already tested values from the solution intervals (##x = -3## and ##x = 6##), but it is equally important to test a value from the excluded middle region. For instance, if we select ##x = 0##, which lies between ##-7/3## and ##5##, we calculate: ### |3(0) – 4| = |-4| = 4 ### Since ##4## is not greater than or equal to ##11##, the value ##x = 0## correctly fails the condition. This corroborates the gap in the solution set and confirms the accuracy of the boundary calculations.

Furthermore, checking the exact boundary points ensures that the equality condition is met. Substituting ##x = 5## into the expression gives ##|3(5) – 4| = |11| = 11##, and substituting ##x = -7/3## gives ##|3(-7/3) – 4| = |-7 – 4| = |-11| = 11##. Both values result in an absolute magnitude exactly equal to the threshold, validating the use of closed intervals. This rigorous multi-step verification process—spanning algebraic checks, region testing, and boundary testing—establishes the robustness of the solution. Geometric mapping provides the necessary intuition to confirm that the algebraic disjunction was handled correctly.

4. Theoretical Deep Dive and Mathematical Context

Axiomatic Properties of Absolute Value Metrics

The behavior of absolute value inequalities is governed by a set of axioms that define the absolute value as a metric in a field. In the field of real numbers ##\mathbb{R}##, the absolute value function ##| \cdot |## satisfies four primary axioms: non-negativity, positive definiteness, multiplicativity, and the triangle inequality. Non-negativity ensures that ##|a| \ge 0## for any ##a##, while positive definiteness states that ##|a| = 0## if and only if ##a = 0##. These axioms are the foundational pillars that allow us to treat the absolute value as a measure of magnitude without regard to direction.

The multiplicativity property, ##|ab| = |a||b|##, allows for the simplification of absolute value expressions involving products. For instance, in our problem, we could have factored out the coefficient ##3## to write ##3|x – 4/3| \ge 11##. The triangle inequality, ##|a + b| \le |a| + |b|##, is perhaps the most famous of these axioms and is vital for proving limits in calculus. These properties demonstrate that the absolute value is more than just a simple “remove the sign” operation; it is a sophisticated mathematical tool used to define distances and topologies in various spaces, including complex numbers and vector spaces.

In a broader context, the absolute value induces a metric ##d(x, y) = |x – y|## which measures the distance between two points on the real line. The inequality ##|3x – 4| \ge 11## can be reinterpreted as a distance-based condition. Specifically, it can be written as ##|x – 4/3| \ge 11/3##, which states that the distance between ##x## and the point ##4/3## must be at least ##11/3##. This metric perspective is essential for understanding more complex inequalities in multi-dimensional spaces, where the absolute value is generalized to the norm of a vector. This theoretical grounding elevates the problem from a simple algebraic exercise to a study in metric geometry.

Historical Foundations of Inequality Formalism

The formalization of the absolute value and the methods for solving its inequalities evolved significantly during the 19th century. The term “absolute value” itself was coined by the French mathematician Augustin-Louis Cauchy in 1806, who used the phrase “valeur absolue” to describe the magnitude of a complex number. However, the modern notation of surrounding an expression with vertical bars ##|x|## was introduced much later, in 1841, by the German mathematician Karl Weierstrass. Before these formalizations, mathematicians often handled magnitude and sign through cumbersome descriptive language rather than symbolic logic.

The development of inequality theory was driven by the need for more rigorous foundations in analysis and calculus. Mathematicians like Jean-Robert Argand and Bernard Bolzano contributed to the early understanding of how magnitudes behave in the context of limits. The shift toward the axiomatic approach in the late 1800s allowed for a standardized method of solving absolute value problems. By the time the modern set-theoretic notation was established in the early 20th century, the procedure for solving disjunctions and conjunctions had become a staple of algebraic education, providing a clear path for students and professionals to navigate complex constraints.

Understanding the history of these concepts provides context for the rigor required in technical problem-solving. The transition from Cauchy’s initial terminology to Weierstrass’s notation reflects a broader trend toward mathematical abstraction. Today, absolute value inequalities are foundational in fields ranging from computer science, where they define error bounds in algorithms, to physics, where they describe physical tolerances. By mastering the algebraic manipulation of these expressions, one engages with a legacy of mathematical thought that spans over two centuries, connecting classical analysis with modern technical implementation. The precision of the notation we use today is a direct result of this historical refinement.