Exercise 41 Page 211

To solve the quadratic inequality algebraically, we will follow three steps.

Solve the related quadratic equation.
Plot the solutions on a number line if exist.
Test a value to see if it satisfies the original inequality.

Step 1

We will start by solving the related equation. - 4x^2+2x= 3 ⇔ - 4x^2+ 2x+( - 3)=0We see above that a= - 4, b= 2, and c= - 3. Let's substitute these values into the Quadratic Formula to solve the equation.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- 2±sqrt(2^2-4( - 4)( - 3))/2( - 4)

▼

Simplify right-hand side

CalcPow

Calculate power

x=- 2±sqrt(4-4(- 4)(- 3))/2(- 4)

Multiply

x=- 2±sqrt(4-48)/- 8

SubTerm

Subtract term

x=- 2±sqrt(- 44)/- 8

We have found a negative discriminant, - 44. Therefore, the related equation has no real solutions.

Step 2

Since the related equation does not have any real solution, we do not have any point to plot on a number line.We have two cases: either all x-values satisfy the original inequalty or no x-value satisfies it.

Step 3

Finally, we must test a value to see if it satisfies the original inequality. Testing one value will help us to determine the solution set. For simplicity, we will choose x=0.

- 4x^2+2x < 3

Substitute

x= 0

- 4( 0)^2+2( 0)? < 3

▼

Simplify left-hand side

CalcPow

Calculate power

- 4(0)+2(0)? <3

ZeroPropMult

Zero Property of Multiplication

0+0 ? < 3