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McGraw Hill Glencoe Algebra 2, 2012

MH

McGraw Hill Glencoe Algebra 2, 2012 View details

8. Quadratic Inequalities

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Exercise 43 Page 286

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

${x ∣ - 2.84 < x < 0.84}$

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.
Test a value from each interval to see if it satisfies the original inequality.

Step $1$

We will start by solving the related equation.

- 12 = - 5 x^{2} - 10 x \Leftrightarrow 5 x^{2} + 10 x + (- 12) = 0

We see above that

a = 5,

b = 10,

and

c = - 12 .

Let's substitute these values into the Quadratic Formula to solve the equation.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

SubstituteValues

Substitute values

x = \frac{- 1 0 \pm 1 0 ^{2} - 4 ( 5 ) ( - 1 2 )}{2 ( 5 )}

▼

Simplify right-hand side

Calculate power

x = \frac{- 1 0 \pm 1 0 0 - 4 ( 5 ) ( - 1 2 )}{2 ( 5 )}

Multiply

x = \frac{- 1 0 \pm 1 0 0 + 2 4 0}{1 0}

Add terms

x = \frac{- 1 0 \pm 3 4 0}{1 0}

Now we can calculate the first root using the positive sign and the second root using the negative sign.

$x = \frac{- 1 0 \pm 3 4 0}{1 0}$
$x = \frac{- 1 0 + 3 4 0}{1 0}$	$x = \frac{- 1 0 - 3 4 0}{1 0}$
$x = - \frac{1 0}{1 0} + \frac{3 4 0}{1 0}$	$x = - \frac{1 0}{1 0} - \frac{3 4 0}{1 0}$
$x \approx 0.84$	$x \approx - 2.84$

Step $2$

The solutions of the related equation are approximately $- 2.84$ and $0.84 .$ Let's plot them on a number line. Since the original is a strict inequality, the points will be open.

Step $3$

Finally, we must test a value from each interval to see if it satisfies the original inequality. Let's choose a value from the first interval,

x < - 2.84 .

For simplicity, we will choose

x = - 4 .

- 12 < - 5 x^{2} - 10 x

$x = - 4$

- 12 < ? - 5 (- 4)^{2} - 10 (- 4)

▼

Simplify left-hand side

Calculate power

- 12 < ? - 5 (16) - 10 (- 4)

Multiply

- 12 < ? - 80 + 40

Add terms

- 12 ≮ - 40 \times

Since

x = - 4

did not produce a true statement, the interval

x < - 2.84

is not part of the solution. Similarly, we can test the other two intervals.

Interval	Test Value	Statement	Is It Part of the Solution?
$- 2.84 < x < 0.84$	$0$	$- 12 < 0$ $✓$	Yes
$x > 0.84$	$1$	$- 12 ≮ - 15$ $\times$	No

We can now write the solution set and show it on a number line.

{x ∣ - 2.84 < x < 0.84} or (- 2.84, 0.84)

Subchapter links

Check Your Understanding p.285

Practice and Problem Solving p.286-287

H.O.T. Problems p.287

Standardized Test Practice p.288

Spiral Review p.288

Skills Review p.288