Solve the equation. 5 x − 2 3 x = 4 + 3 x \frac{5}{x}-\frac{2}{3x}=4+\frac{3}{x}

Hey, everyone. Let's take a look at this practice problem. So here I have 5 over x minus 2 over 3 x is equal to 4 +3x, And I wanna solve this rational equation. So let's go ahead and get started with step 1, which is to determine our restriction. Now my 3 denominators I have are x, 3x, and x. So by simply looking at these, I can see that if I were to plug in x equals 0 for any of them, that would make my denominator 0. So I know that x cannot be 0, and that is my restriction. So step 1 is done. I can go ahead and move on to step 2, which is to multiply by my least common denominator, my LCD, to eliminate fractions. Now, again, my 3 denominators are x, 3x, and x, and they all have the common factor of x. And then I have that extra 3 there, so my least common denominator is going to be 3x. Now remember when we multiply by our least common denominator, we want to make sure to distribute to every single term in our equation. So let's go ahead and expand this out. So I have 3x times 5 over x minus 3x times 2 over 3x. And that's equal to 3x times 4. And then finally, my last term, my least common denominator, 3x times 3 over x. Now it looks a little crazy right now. Let's go ahead and simplify a little bit. So in my very first term, this x is going to cancel, leaving me with 3 times 5, which is just 15. And then I have this 3x canceling here, leaving me with simply minus 2. And that's equal to 3x times 4, which just gives me positive 12 x. And then lastly, this x and x are going to cancel. I'm left with 3 times 3, which is simply a positive 9. So step 2 is done. Let's move on to step 3, solving the linear equation that we're left with. So I don't have anything that needs to get distributed, but I do have a like term right here, 15 and negative 2. So this will simplify to 13. I don't have any other like terms I need to get combined, so I just have 12 x plus 9. Now we wanna move all our x terms to one side, all of our constants to the other. Since I only have one x term and it's already on that right side, I'm going to keep it there and move my 9 over by subtracting it. So it will, of course, cancel there. And I'm left with 13 minuteus 9, which will give me a positive 4, and that's equal to 12 x. My last step is, of course, to isolate my x, which I can do by dividing both sides by 12. Now I can actually simplify that a little bit further, simplify that fraction 4 over 12. This simplifies to 1 third, and that's equal to x. So this is my solution, x is equal to 1 third. We can, of course, write it the other way if that makes you happy. So I have x equals 1 3rd. I've completed step 3. Now step 4 is, of course, to check my solution with my restriction. Now my restriction was that x could not be equal to 0, and x is not equal to 0. It's equal to 1 third, and that is my solution. That's all for this one. I'll see you in the next one.

Skip topic navigation

1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

12. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

13. Sequences, Series, and the Binomial Theorem1h 50m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

8. Rational Expressions and Functions

Rational Equations

Intermediate Algebra

8. Rational Expressions and Functions

Rational Equations

이전 문제다음 문제

Intermediate Algebra 때문에 힘들어하고 계신가요?

저희를 신뢰하는 수천 명의 학생들이 시험 합류하세요!

객관식

Solve the equation.
$\frac{5}{x} - \frac{2}{3 x} = 4 + \frac{3}{x}$

$x=0$

$x=1$

$x=$\frac$13$

No solution

0 댓글

이전 문제다음 문제

검증된 단계별 안내

Start with the given equation: $\frac{5}{x} - \frac{2}{3x} = 4 + \frac{3}{x}$.

Identify the least common denominator (LCD) for all the fractions, which is \$3x$.

Multiply every term in the equation by the LCD \$3x$ to eliminate the denominators: $3x \times \frac{5}{x} - 3x \times \frac{2}{3x} = 3x \times 4 + 3x \times \frac{3}{x}$.

Simplify each term after multiplication: $3x \times \frac{5}{x} = 15$, $3x \times \frac{2}{3x} = 2$, $3x \times 4 = 12x$, and $3x \times \frac{3}{x} = 9$.

Rewrite the equation without fractions: $15 - 2 = 12x + 9$. Then, combine like terms and solve the resulting linear equation for $x$.