\( |6 t-5| \leq 1 \) Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. (Use integers or fractions for any numbers in the expression. Simplify your answer.) A. The solution set is \( \{t \mid \square \leq t \leq \square\} \) B. The solution set is \( \{t \mid t \leq \square \) or \( t \geq \square\} \). C. The solution set is the set of all real numbers. D. The solution set is the empty set.

To solve the inequality \( |6t - 5| \leq 1 \), we start by breaking it down into two parts: 1. \( 6t - 5 \leq 1 \) 2. \( 6t - 5 \geq -1 \) Solving these gives: 1. For \( 6t - 5 \leq 1 \): \[ 6t \leq 6 \] \[ t \leq 1 \] 2. For \( 6t - 5 \geq -1 \): \[ 6t \geq 4 \] \[ t \geq \frac{2}{3} \] Thus, the combined solution is: \[ \frac{2}{3} \leq t \leq 1 \] So the correct choice is: A. The solution set is \( \{t \mid \frac{2}{3} \leq t \leq 1\} \) --- The absolute value is a mathematical concept with roots tracing back to the early 19th century when mathematicians began to emphasize the need for a measure of distance on the number line. It helps in expressing and solving inequalities by defining how far a number is from zero without considering direction. Using absolute value inequalities like this allows us to model real-world situations that involve tolerances or limits, such as engineering specifications where measurements can deviate by a certain amount. For example, if a product is supposed to measure 5 units with an acceptable variation of ±1 unit, you would set up a similar inequality to find acceptable manufacturing ranges!