The set of values of x satisfying the inequality |x-1|+|x-2| ≥6 is (a) (-∞, (-3)/(2)] ∪[(9)/(2),

step 1 Okay, so in this question, we are given an expression in mode, which is mod X minus 1 plus mode

The set of values of x satisfying the inequality |x-1|+|x-2|...

The set of values of $\mathrm{x}$ satisfying the inequality $|\mathrm{x}-1|+|\mathrm{x}-2| \geq 6$ is
(a) $\left(-\infty, \frac{-3}{2}\right] \cup\left[\frac{9}{2}, \infty\right)$
(b) $\left(\frac{-3}{2}, \frac{9}{2}\right)$
(c) $\left(-\infty, \frac{3}{2}\right]$
(d) $(1,2)$

Key Concepts

Solving Absolute Value Inequalities

Solving inequalities that involve absolute values typically involves considering different cases based on the piecewise definitions. By splitting the problem into intervals determined by the critical points, one can convert the absolute value inequality into one or more standard inequalities, which can then be solved using usual algebraic methods.

Critical Points

Critical points are the values where the expressions inside the absolute value become zero. These points, often called breakpoints, partition the number line into intervals where the behavior of the absolute value function is consistent. They are crucial for setting up the correct cases when analyzing and solving absolute value inequalities.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. In algebra, it is used to express the magnitude of a difference and is defined piecewise, which is essential when dealing with equations or inequalities that contain absolute values.

Piecewise Definition

Absolute value functions are defined in a piecewise manner, meaning that the expression inside the absolute value is considered separately based on whether it is non-negative or negative. This approach allows breaking a problem into cases and simplifies the process of solving equations or inequalities involving absolute values.

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