Solve each inequality). 2|4t-1|+6>20

Question

asked Oct 12, 2023 230k views

1 Answer

Hivaga · Answer 1 · 2023-10-18T22:36:32+0000

To answer this question we will use the following property:

$|a|>b>0\text{ if and only if }a>b\text{ or }a<-b.$

Subtracting 6 from the given inequality we get:

$\begin{gathered} 2|4t-1|+6-6>20-6, \\ 2|4t-1|>14. \end{gathered}$

Dividing the above inequality by 2 we get:

$\begin{gathered} (2|4t-1|)/(2)>(14)/(2), \\ |4t-1|>7. \end{gathered}$

Then:

$4t-1>7\text{ or }4t-1<-7.$

Solving the above inequalities we get:

1)

Adding 1 to the above inequality we get:

$\begin{gathered} 4t-1+1>7+1, \\ 4t>8. \end{gathered}$

Dividing the above by 4 we get:

$\begin{gathered} (4t)/(4)>(8)/(4), \\ t>2. \end{gathered}$

The above inequality in interval notation is:

$(2,\infty).$

2)

Adding 1 to the above inequality we get:

$\begin{gathered} 4t-1+1<-7+1, \\ 4t<-6. \end{gathered}$

Dividing the above result by 4 we get:

$\begin{gathered} (4t)/(4)<-(6)/(4), \\ t<-(3)/(2). \end{gathered}$

The above inequality in interval notation is:

$(-\infty,-(3)/(2)).$

Answer:

$(-\infty,-(3)/(2))\cup(2,\infty).$