1. **Problem Statement:** Find the absolute value of given numbers and expressions, evaluate absolute value expressions, and solve absolute value equations. 2. **Formula and Rules:** - Absolute value of a number $a$ is defined as: $$|a| = \begin{cases} a, & \text{if } a \geq 0 \\ -a, & \text{if } a < 0 \end{cases}$$ - Absolute value represents the distance from zero on the number line, so it is always non-negative. - For equations like $|x - a| = b$, solutions are $x = a + b$ or $x = a - b$ if $b \geq 0$. - If the right side of an absolute value equation is negative, there is no solution. 3. **Step-by-step Solutions:** **1a) Find $|6|$** - Since $6 \geq 0$, $|6| = 6$. **2a) Evaluate $|x - 4|$** - By definition, $|x - 4| = \begin{cases} x - 4, & x \geq 4 \\ 4 - x, & x < 4 \end{cases}$. **1d) Find $|1 - \sqrt{3}|$** - Calculate $1 - \sqrt{3} \approx 1 - 1.732 = -0.732 < 0$. - So, $|1 - \sqrt{3}| = -(1 - \sqrt{3}) = \sqrt{3} - 1$. **3e) Solve $|x - 1| = \pi$** - Since $\pi > 0$, solutions are: $$x - 1 = \pi \quad \text{or} \quad x - 1 = -\pi$$ - Thus, $$x = \pi + 1 \quad \text{or} \quad x = 1 - \pi$$ - Solution set: $\{\pi + 1, 1 - \pi\}$. **Additional notes:** - For equations like $|x| = -1$, no solution exists because absolute value cannot be negative. 4. **Summary:** - Absolute value outputs non-negative values. - Equations with negative right sides have no solutions. - Equations $|expression| = positive$ have two solutions. Final answer for the first problem (1a): $$|6| = 6$$