3/2x-2+1/2=2/x-1

This solution deals with adding, subtracting và finding the least common multiple.

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Step by Step Solution

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Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 3/2*x-2+1/2-(2/x-1)=0

Step by step solution :

Step 1 :

2 Simplify — xEquation at the end of step 1 : 3 1 2 (((—•x)-2)+—)-(—-1) = 0 2 2 x

Step 2 :

Rewriting the whole as an Equivalent Fraction :2.1Subtracting a whole from a fraction Rewrite the whole as a fraction using x as the denominator :

1 1 • x 1 = — = ————— 1 x Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction và the other fraction involved in the calculation mô tả the same denominator

Adding fractions that have a common denominator :2.2 Adding up the two equivalent fractions địa chỉ the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

2 - (x) 2 - x ——————— = ————— x x Equation at the end of step 2 : 3 1 (2-x) (((—•x)-2)+—)-————— = 0 2 2 x

Step 3 :

1 Simplify — 2Equation at the end of step 3 : 3 1 (2 - x) (((— • x) - 2) + —) - ——————— = 0 2 2 x

Step 4 :

3 Simplify — 2Equation at the kết thúc of step 4 : 3 1 (2 - x) (((— • x) - 2) + —) - ——————— = 0 2 2 x

Step 5 :

Rewriting the whole as an Equivalent Fraction :5.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 2 as the denominator :

2 2 • 2 2 = — = ————— 1 2 Adding fractions that have a common denominator :5.2 Adding up the two equivalent fractions

3x - (2 • 2) 3x - 4 ———————————— = —————— 2 2 Equation at the kết thúc of step 5 : (3x - 4) 1 (2 - x) (———————— + —) - ——————— = 0 2 2 x

Step 6 :

Adding fractions which have a common denominator :6.1 Adding fractions which have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce khổng lồ lowest terms if possible:

(3x-4) + 1 3x - 3 —————————— = —————— 2 2 Equation at the over of step 6 : (3x - 3) (2 - x) ———————— - ——————— = 0 2 x

Step 7 :

Step 8 :

Pulling out lượt thích terms :8.1 Pull out lượt thích factors:3x - 3=3•(x - 1)

Calculating the Least Common Multiple :8.2 Find the Least Common Multiple The left denominator is : 2 The right denominator is : x

Number of times each prime factorappears in the factorization of:PrimeFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right
2101
Product of allPrime Factors212

Number of times each Algebraic Factorappears in the factorization of:AlgebraicFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right
x011

Least Common Multiple: 2x

Calculating Multipliers :

8.3 Calculate multipliers for the two fractions Denote the Least Common Multiple by L.C.M Denote the Left Multiplier by Left_M Denote the Right Multiplier by Right_M Denote the Left Deniminator by L_Deno Denote the Right Multiplier by R_DenoLeft_M=L.C.M/L_Deno=xRight_M=L.C.M/R_Deno=2

Making Equivalent Fractions :

8.4 Rewrite the two fractions into equivalent fractions

L. Mult. • L. Num. 3 • (x-1) • x —————————————————— = ————————————— L.C.M 2x R. Mult. • R. Num. (2-x) • 2 —————————————————— = ————————— L.C.M 2x Adding fractions that have a common denominator :8.5 Adding up the two equivalent fractions

3 • (x-1) • x - ((2-x) • 2) 3x2 - x - 4 ——————————————————————————— = ——————————— 2x 2x Trying to factor by splitting the middle term8.6Factoring 3x2 - x - 4 The first term is, 3x2 its coefficient is 3.The middle term is, -x its coefficient is -1.The last term, "the constant", is -4Step-1 : Multiply the coefficient of the first term by the constant 3•-4=-12Step-2 : Find two factors of -12 whose sum equals the coefficient of the middle term, which is -1.

-12+1=-11
-6+2=-4
-4+3=-1That"s it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 và 33x2 - 4x+3x - 4Step-4 : địa chỉ up the first 2 terms, pulling out lượt thích factors:x•(3x-4) add up the last 2 terms, pulling out common factors:1•(3x-4) Step-5:Add up the four terms of step4:(x+1)•(3x-4)Which is the desired factorization

Equation at the end of step 8 : (3x - 4) • (x + 1) —————————————————— = 0 2x

Step 9 :

When a fraction equals zero :9.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.Here"s how:

(3x-4)•(x+1) ———————————— • 2x = 0 • 2x 2x Now, on the left hand side, the 2x cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:(3x-4) • (x+1)=0

Theory - Roots of a product :9.2 A hàng hóa of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to lớn solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.

Solving a Single Variable Equation:9.3Solve:3x-4 = 0Add 4 to lớn both sides of the equation:3x = 4 Divide both sides of the equation by 3:x = 4/3 = 1.333

Solving a Single Variable Equation:9.4Solve:x+1 = 0Subtract 1 from both sides of the equation:x = -1

Supplement : Solving Quadratic Equation Directly

Solving 3x2-x-4 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex:10.1Find the Vertex ofy = 3x2-x-4Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,3, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want lớn be able khổng lồ find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is 0.1667Plugging into the parabola formula 0.1667 for x we can calculate the y-coordinate:y = 3.0 * 0.17 * 0.17 - 1.0 * 0.17 - 4.0 or y = -4.083

Parabola, Graphing Vertex & X-Intercepts :Root plot for : y = 3x2-x-4 Axis of Symmetry (dashed) x= 0.17 Vertex at x,y = 0.17,-4.08 x-Intercepts (Roots) : Root 1 at x,y = -1.00, 0.00 Root 2 at x,y = 1.33, 0.00

Solve Quadratic Equation by Completing The Square

10.2Solving3x2-x-4 = 0 by Completing The Square.Divide both sides of the equation by 3 lớn have 1 as the coefficient of the first term :x2-(1/3)x-(4/3) = 0Add 4/3 khổng lồ both side of the equation : x2-(1/3)x = 4/3Now the clever bit: Take the coefficient of x, which is 1/3, divide by two, giving 1/6, & finally square it giving 1/36Add 1/36 to lớn both sides of the equation :On the right hand side we have:4/3+1/36The common denominator of the two fractions is 36Adding (48/36)+(1/36) gives 49/36So adding lớn both sides we finally get:x2-(1/3)x+(1/36) = 49/36Adding 1/36 has completed the left hand side into a perfect square :x2-(1/3)x+(1/36)=(x-(1/6))•(x-(1/6))=(x-(1/6))2 Things which are equal khổng lồ the same thing are also equal khổng lồ one another. Sincex2-(1/3)x+(1/36) = 49/36 andx2-(1/3)x+(1/36) = (x-(1/6))2 then, according lớn the law of transitivity,(x-(1/6))2 = 49/36We"ll refer to lớn this Equation as Eq.

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#10.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(1/6))2 is(x-(1/6))2/2=(x-(1/6))1=x-(1/6)Now, applying the Square Root Principle lớn Eq.#10.2.1 we get:x-(1/6)= √ 49/36 add 1/6 lớn both sides khổng lồ obtain:x = 1/6 + √ 49/36 Since a square root has two values, one positive & the other negativex2 - (1/3)x - (4/3) = 0has two solutions:x = 1/6 + √ 49/36 orx = 1/6 - √ 49/36 cảnh báo that √ 49/36 can be written as√49 / √36which is 7 / 6

Solve Quadratic Equation using the Quadratic Formula

10.3Solving3x2-x-4 = 0 by the Quadratic Formula.According to lớn the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B & C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 3B= -1C= -4 Accordingly,B2-4AC=1 - (-48) = 49Applying the quadratic formula : 1 ± √ 49 x=—————6Can √ 49 be simplified ?Yes!The prime factorization of 49is7•7 to lớn be able khổng lồ remove something from under the radical, there have khổng lồ be 2 instances of it (because we are taking a square i.e. Second root).√ 49 =√7•7 =±7 •√ 1 =±7 So now we are looking at:x=(1±7)/6Two real solutions:x =(1+√49)/6=(1+7)/6= 1.333 or:x =(1-√49)/6=(1-7)/6= -1.000