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Math  /  Algebra

QuestionQuiz Question 8 of 11 This quiz: 11 This question
Solve the following equation analytically. Check it analytically by direct substitution, and then support your solution graphically. 2x+23+x−24=152\frac{2 x+2}{3}+\frac{x-2}{4}=\frac{15}{2}
The solution set of this equation is □\square (Simplify your answer.)

Studdy Solution

STEP 1

1. The equation involves fractions and requires finding a common denominator.
2. The solution involves solving a linear equation.
3. Verification involves substituting the solution back into the original equation.
4. Graphical support involves plotting the equation and identifying the solution visually.

STEP 2

1. Simplify the equation by finding a common denominator.
2. Solve the simplified linear equation for x x .
3. Verify the solution by direct substitution.
4. Support the solution graphically.

STEP 3

First, identify the least common denominator (LCD) for the fractions. The denominators are 3, 4, and 2. The LCD is 12.
Rewrite each term with the LCD as the denominator:
2x+23=4(2x+2)12=8x+812 \frac{2x + 2}{3} = \frac{4(2x + 2)}{12} = \frac{8x + 8}{12}
x−24=3(x−2)12=3x−612 \frac{x - 2}{4} = \frac{3(x - 2)}{12} = \frac{3x - 6}{12}
152=6×1512=9012 \frac{15}{2} = \frac{6 \times 15}{12} = \frac{90}{12}
Combine the fractions:
8x+812+3x−612=9012 \frac{8x + 8}{12} + \frac{3x - 6}{12} = \frac{90}{12}
Combine the numerators:
(8x+8)+(3x−6)12=9012 \frac{(8x + 8) + (3x - 6)}{12} = \frac{90}{12}
11x+212=9012 \frac{11x + 2}{12} = \frac{90}{12}

STEP 4

Since the denominators are the same, equate the numerators:
11x+2=90 11x + 2 = 90
Subtract 2 from both sides:
11x=88 11x = 88
Divide by 11:
x=8 x = 8

STEP 5

Verify the solution by substituting x=8 x = 8 back into the original equation:
2(8)+23+8−24=152 \frac{2(8) + 2}{3} + \frac{8 - 2}{4} = \frac{15}{2}
Calculate each term:
16+23=183=6 \frac{16 + 2}{3} = \frac{18}{3} = 6
8−24=64=32 \frac{8 - 2}{4} = \frac{6}{4} = \frac{3}{2}
Add the results:
6+32=122+32=152 6 + \frac{3}{2} = \frac{12}{2} + \frac{3}{2} = \frac{15}{2}
The left side equals the right side, confirming the solution is correct.

STEP 6

Support the solution graphically by plotting the left side and right side of the original equation as functions of x x :
Plot y1=2x+23+x−24 y_1 = \frac{2x + 2}{3} + \frac{x - 2}{4} and y2=152 y_2 = \frac{15}{2} .
The solution x=8 x = 8 is where the graphs of y1 y_1 and y2 y_2 intersect.
The solution set of this equation is:
8 \boxed{8}

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