Chapter 1.9, Problem 29E

Solution

To calculate: The value of x for the inequality $x^2-2x\ge 8$ using a table.

The solution of the inequality is $x\le -2\text{and}x\ge 4$ .

Given information:

The inequality is $x^2-2x\ge 8$ .

Formula used:

Substitute the values for x into the inequality.

Calculation:

Write the given inequality:

  $x^2-2x\ge 8$

Subtract 8 from both sides to make the right side of the inequality equal to 0,

  $\begin{array}{l}{x}^2-2x-8\ge 8-8\\ x^2-2x-8\ge 0\end{array}$

Substitute the $x=-4$ in the inequality,

  $\begin{array}{c}{\left(-4\right)}^2-2\left(-4\right)-8\ge 0\\ 16+8-8\ge 0\\ 16\ge 0\end{array}$

Substitute the $x=-3$ in the inequality,

  $\begin{array}{c}{\left(-3\right)}^2-2\left(-3\right)-8\ge 0\\ 9+6-8\ge 0\\ 7\ge 0\end{array}$

Substitute the $x=-2$ in the inequality,

  $\begin{array}{c}{\left(-2\right)}^2-2\left(-2\right)-8\ge 0\\ 4+4-8>0\\ 0=0\end{array}$

Similarly make a table by substituting the values of x in the inequality:

x	$-4$	$-3$	$-2$	$-1$	0	1	2	3	4	5
$x^2-2x-8$	16	7	0	$-5$	$-8$	$-9$	$-8$	$-5$	0	7

It can be seen that $x^2-2x-8\ge 0$ if values of x are taken less than or equal to $-2$ and more than or equal to 4 are substituted in the inequality.

Therefore, $x\le -2\text{and}x\ge 4$ are the solutions for the inequality $x^2-2x-8\ge 0$ .