Question

What are the minimum and maximum values of t for which |t + 3| ≤ 2? minimum: maximum:

Answer 1

It + 3I is called absolut value which means the value of t + 3 is always positive

So we will solve two inequalities

$\begin{gathered} t+3\leq2 \\ -(t+3)\leq2 \end{gathered}$

Let us solve the first one

Subtract from both sides 3

$t+3-3\leq2-3$$t\leq-1$

Now we will solve the second one

We multiply the bracket by (-)

$-t-3\leq2$

Add 3 to both sides

$\begin{gathered} -t-3+3\leq2+3 \\ -t\leq5 \end{gathered}$

We will divide both sides by -1 the coefficient of t, but we must reverse the sign of the inequality because 2 < 3 if we divide both sides by -1 it will be -2 < -3 which is wrong -2 > -3, so

When you multiply or divide an inequality by negative number you must reverse the sign of the inequality

$\begin{gathered} -1(-t)\ge-1(5) \\ t\ge-5 \end{gathered}$

Now we will write the both inequalities togather

$-5\leq t\leq-1$

The minimum value is -5

The maximum value is -1