Question

Solve each inequality algebraically.

$7x-4\ge -2x^2$

Short answer

Solution set and the interval is:

$$\begin{gathered} \left\{x\right|x\le -4orx\ge \frac{1}{2}\left\} \\ \right(-\infty ,-4]\cup [\frac{1}{2},\infty ) \end{gathered}$$

Step-by-step solution

Step 1. Change the inequality to equal to zero. 

Change the inequality equal to zero to make it easier and then get the value of x.

$$\begin{gathered} 7x-4=-2x^2 \\ 2x^2+7x-4=0 \\ (2x-1)(x+4)=0 \\ 2x-1=0;x+4=0 \\ x=\frac{1}{2};x=-4 \end{gathered}$$

Step 2. Form the intervals

From the obtained values of x, we can form the interval,

So the interval we have is:

$(-\infty ,-4)\cup (-4,\frac{1}{2})\cup (\frac{1}{2},\infty )$

Step 3. Form the table

Since, $f\left(x\right)\ge 0$, so is positive or zero.

check a number in each interval and evaluate the function to see whether it is satisfying the function.

Interval	Number chosen	Resultant
$(-\infty ,-4)$	-6	$24⩾0$
$(-4,\frac{1}{2})$	0	$-4\ge 0$
$(\frac{1}{2},\infty )$	1	$5\ge 0$

Step 4. Solution set and interval

Since,

$24\ge 0$and $5\ge 0$satisfy $f$. Thus,

role="math" localid="1646158628522" $$\begin{gathered} \left\{x\right|x\le -4orx\ge \frac{1}{2}\left\} \\ \right(-\infty ,-4]\cup [\frac{1}{2},\infty ) \end{gathered}$$