Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 23

Chapter 2, Problem 23

Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(x + 8) - √(x - 4) = 2

Verified step by step guidance

Start by isolating one of the square root terms. For example, rewrite the equation as $\sqrt{{x + 8}} = \sqrt{{x - 4}} + 2$.

Square both sides of the equation to eliminate the square root on the left. This gives: $\left(\sqrt{{x + 8}}\right)^2 = \left(\sqrt{{x - 4}} + 2\right)^2$.

Simplify both sides: the left side becomes $x + 8$, and the right side expands using the formula $(a + b)^2 = a^2 + 2ab + b^2$ to $ (x - 4) + 4\sqrt{{x - 4}} + 4$.

Rearrange the equation to isolate the remaining square root term: $x + 8 = x - 4 + 4\sqrt{{x - 4}} + 4$, then simplify to get $4\sqrt{{x - 4}} = 8$.

Divide both sides by 4 to isolate the square root: $\sqrt{{x - 4}} = 2$. Then square both sides again to solve for $x$: $x - 4 = 4$. Finally, solve for $x$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radical Equations

Radical equations involve variables inside a root, often a square root. To solve them, isolate the radical expression and then eliminate the root by raising both sides of the equation to the appropriate power, typically squaring for square roots.

Domain Restrictions

When dealing with radicals, especially square roots, the expression inside the root must be non-negative to produce real solutions. Identifying domain restrictions ensures that any proposed solutions are valid within the problem's context.

Checking Solutions

After solving radical equations, it is essential to substitute solutions back into the original equation to verify they do not produce extraneous results. Squaring both sides can introduce invalid solutions, so checking prevents incorrect answers.

Recommended video:

05:21

Restrictions on Rational Equations

Related Practice

Textbook Question

Solve each equation in Exercises 15–34 by the square root property. $3 (x - 4)^{2} = 15$

views

Textbook Question

Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3

y = -2|x|

Textbook Question

You invested \$30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was \$705.88, how much was invested at each rate?

Textbook Question

In Exercises 1–26, solve and check each linear equation. 2 - (7x + 5) = 13 - 3x

views

Textbook Question

In Exercises 15–26, use graphs to find each set. [3, ∞) ∩ (6, ∞)

views

Textbook Question

In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)

views