Chapter 5.8, Problem 6WE

To determine

To find the open sentence of $\frac{t}{4}-\frac{t+2}{3}+\frac{1}{6}\ge 0$

Answer to Problem 6WE

The value of open sentence is $t\le -6$

Explanation of Solution

Given open sentence is $\frac{t}{4}-\frac{t+2}{3}+\frac{1}{6}\ge 0$

Value of open sentence means a sentence is it true and depend upon the variable value .

  $\frac{t}{4}-\frac{t+2}{3}+\frac{1}{6}\ge 0$

Multiply both the sides by 12, the LCM of 3, 4 and 6

  $12\left(\frac{t}{4}-\frac{t+2}{3}+\frac{1}{6}\right)\ge (12)0$

Apply distributive property and simplify $\left[a(b\pm c)=a\ast b\pm a*c\right]$

  $\begin{array}{c}3t-4(t+2)+2\ge 0\\ 3t-4t-8+2\ge 0\\ -t-6\ge 0\end{array}$

Add 6 to both the sides

  $\begin{array}{c}-t-6+6\ge 6\\ -t\ge 6\end{array}$

Now, reverse the inequality

  $t\le -6$

Hence, the open sentence value is $t\le -6$