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Chapter 7: Q. 366 (page 732)

In the following exercises, solve each rational inequality and write the solution in interval notation.

12-32x2鈮�1x

Short Answer

Expert verified

Solution for rational inequality isx鈭�(-鈭�,1]鈭�[3,鈭�).

Step by step solution

01

Step 1. Given Information

Given rational inequality is12-32x2鈮�1x.

02

Step 2. Definition of inequality

On subtracting 1xon both sides,

12-32x2-1x鈮�1x-1x12-32x2-1x鈮�0x2-3-2x2x2鈮�0

For the definition of inequality, denominator should not be zero. Thus, one of the critical points isx=0.

03

Step 3. Required condition

To true the above inequality, the condition can be stated as,

x2-3-2x鈮�0f(x)=x2-2x-3

On factorizing the polynomial using AC method.

ax2+bx+cx2-3-2x鈮�0x2+x-3x-3鈮�0x(x+1)-3(x+1)鈮�0(x+1)(x+3)鈮�0

04

Step 4. Critical points 

To find the critical points,

f(x)=0x2-2x-3=0(x+1)(x-3)=0x=-1,3

Thus, the critical points arex=0,-1,3.

05

Step 5. Testing of critical points

To test these critical points, value of the x2-3-2x2x2at different points.

x=-2is4-3+48=58x=0.5is0.25-3-10.5=-7.5x=2is4-3-48=-38x=4is16-3-832=532

This satisfies the condition ofx鈭�(-鈭�,-1]鈭�[3,鈭�).

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