body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

5. Quadratic Inequalities

Continue to next subchapter

Exercise 50 Page 212

Solve the related quadratic equation, plot the solutions on a number line, and test a value from each interval.

{all real numbers }

Practice makes perfect

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. 5x^2+ 1x+ 3=0We see above that a= 5, b= 1, 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=- 1±sqrt(1^2-4( 5)( 3))/2( 5)

▼

Simplify right-hand side

1^a=1

x=- 1±sqrt(1-4(5)(3))/2(5)

Multiply

x=- 1±sqrt(1-60)/10

Subtract term

x=- 1±sqrt(- 59)/10

We have found a negative discriminant, - 59. 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 inequality 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.

5x^2+x+3 ≥ 0

x= 0

5( 0)^2+ 0+3? ≥ 0

▼

Simplify left-hand side

Calculate power

5(0)+0+3? ≥ 0

Zero Property of Multiplication

0+0+3? ≥ 0

Identity Property of Addition

3 ≥ 0 ✓

Since x=0 produced a true statement, all x-values satisfy the inequality. We can now write the solution set. {all real numbers } or (- ∞, ∞)

Subchapter links

Exercises p.210-213