Solve each equation. Check your solutions. See Example 1 ....

Key Concepts

Rational Equations

Rational equations involve expressions that contain variables in the denominators. Solving these equations often requires working with fractions and managing expressions to combine or eliminate denominators, making the equation easier to solve.

Domain Restrictions

When working with rational expressions, it is crucial to determine the values for the variable that would make any denominator zero. These restrictions are necessary to avoid undefined expressions and to ensure that any solutions obtained are valid in the context of the original equation.

Clearing Fractions

A common technique in solving rational equations is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD). This step transforms the rational equation into a polynomial equation, which is typically simpler to solve.

Extraneous Solutions and Verification

Operations used to clear fractions and simplify the equation can sometimes introduce solutions that do not satisfy the original equation, known as extraneous solutions. It is important to substitute the solutions back into the original equation to verify their validity and ensure that none of the domain restrictions are violated.

Transcript

00:01 So we have x over x minus 2 plus 2 over x is equal to 5.

00:06 Okay, so while looking for a common denominator here is going to be, well, 2 minus x times x.

00:24 Okay, or in other words, x squared minus or 2x, 2x minus 2x minus.

00:31 X squared is going to be a common denominator.

00:34 So we can just multiply both sides by that common denominator.

00:38 So we get our left side, we get x over 2 minus x plus 2 over x.

00:48 Okay.

00:48 And then we're going to multiply this left side by, well, by 2x minus x squared.

00:56 Okay.

00:57 And then do the same thing to the right side.

01:00 This is equal to 5.

01:02 And then we're multiplying by that same lcd, multiplying by 2x minus x squared.

01:08 So multiplying by 2x minus x squared.

01:11 Okay, so multiplying both sides, but the same thing, so i'm not changing the, so i'm keeping the equation equal.

01:17 Okay, then, well, let's see, what do we get here? well, we're gonna get something's cancelling out.

01:32 We end up with getting x squared, x squared plus four minus two x um is equal to 10x um right in the five distributes you get equal to 10x minus 5x squared so minus 5 x squared okay then we're going to put this thing in standard form so um we want to basically um get all of our you know well equal to zero and then combine like terms and we end up with in lowest terms so we get a let's see we end up with 3 x squared minus 6x plus 2 is equal to 0 okay so now well this is in um i was going to factor really nice but we can we have this in standard form set equal to so we could always use the quadratic equation to solve this.

02:48 So we have that while x is equal to well negative b so negative a negative 6 so positive 6 plus or minus the square root of b squared.

03:02 So negative 6 squared is a positive 36 and then minus 4 times ac so minus 4 times 3 times 2...

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