How can you solve this quadratic inequality by using the interval method?2(xsquared) - 9 > 23 thanks

1 Answer | Add Yours

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Solve `2x^2-9>23` .

(1) First we rewrite as an equivalent inequality. (An equivalent inequality has the same solution set)

`2x^2-32>0`

(2) Factor the left-hand side as `2(x^2-16)>0` or `2(x+4)(x-4)>0` .

(3) Note that the zeros occur at x=-4 and x=4. Then we divide the x-axis into 3 intervals: `(-oo,-4),(-4,4),(4,oo)` .

(4) Choose a point in each interval and compute the value of the function.

`f(-5)=18>0`
`f(0)=-32<0`
`f(5)=18>0`

(5) Then the solution set is `(-oo,-4)uu(4,oo)` or `x<-4,x>4`

As a check, the graph:

` <img style="width: 300px; height: 200px; vertical-align: middle; float: none;" _mce_style="width: 300px; height: 200px; vertical-align: middle; float: none;" src="http://www.enotes.com/util/tinymce_math/svgimg.php?sscr=-7.5%2C7.5%2C-5%2C30%2C1%2C1%2C1%2C1%2C1%2C300%2C200%2Cfunc%2C2x%5E2-9%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone%2Cfunc%2C23%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone" _mce_src="http://www.enotes.com/util/tinymce_math/svgimg.php?sscr=-7.5%2C7.5%2C-5%2C30%2C1%2C1%2C1%2C1%2C1%2C300%2C200%2Cfunc%2C2x%5E2-9%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone%2Cfunc%2C23%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone" sscr="-7.5,7.5,-5,30,1,1,1,1,1,300,200,func,2x^2-9,null,0,0,,,black,1,none,func,23,null,0,0,,,black,1,none" script=" "> `

Note that the graph is above the line y=23 to the left of -4, and to the right of 4 as found above.

We’ve answered 318,916 questions. We can answer yours, too.

Ask a question