Watkins Wade
04/02/2023 · Elementary School
10. Solve the following equations. (a) \( 3(2 x+1)-2(2 x-7)=5(x-11) \) (b) \( \left(\frac{1}{x}-2\right)^{2}=\frac{1}{9} \) (c) \( \frac{2 x-1}{4}-\frac{x+1}{3}+1=0 \)
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Step-by-step Solution
Solve the equation \( 3(2x+1)-2(2x-7)=5(x-11) \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3\left(2x+1\right)-2\left(2x-7\right)=5\left(x-11\right)\)
- step1: Calculate:
\(2x+17=5\left(x-11\right)\)
- step2: Calculate:
\(2x+17=5x-55\)
- step3: Move the expression to the left side:
\(2x+17-\left(5x-55\right)=0\)
- step4: Calculate:
\(-3x+72=0\)
- step5: Move the constant to the right side:
\(-3x=0-72\)
- step6: Remove 0:
\(-3x=-72\)
- step7: Change the signs:
\(3x=72\)
- step8: Divide both sides:
\(\frac{3x}{3}=\frac{72}{3}\)
- step9: Divide the numbers:
\(x=24\)
Solve the equation \( \left(\frac{1}{x}-2\right)^{2}=\frac{1}{9} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\left(\frac{1}{x}-2\right)^{2}=\frac{1}{9}\)
- step1: Find the domain:
\(\left(\frac{1}{x}-2\right)^{2}=\frac{1}{9},x\neq 0\)
- step2: Simplify the expression:
\(\frac{1}{x}-2=\pm \sqrt{\frac{1}{9}}\)
- step3: Simplify:
\(\frac{1}{x}-2=\pm \frac{1}{3}\)
- step4: Separate into possible cases:
\(\begin{align}&\frac{1}{x}-2=\frac{1}{3}\\&\frac{1}{x}-2=-\frac{1}{3}\end{align}\)
- step5: Calculate:
\(\begin{align}&x=\frac{3}{7}\\&x=\frac{3}{5}\end{align}\)
- step6: Check if the solution is in the defined range:
\(\begin{align}&x=\frac{3}{7}\\&x=\frac{3}{5}\end{align},x\neq 0\)
- step7: Find the intersection:
\(\begin{align}&x=\frac{3}{7}\\&x=\frac{3}{5}\end{align}\)
- step8: Rewrite:
\(x_{1}=\frac{3}{7},x_{2}=\frac{3}{5}\)
Solve the equation \( \frac{2x-1}{4}-\frac{x+1}{3}+1=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{2x-1}{4}-\frac{x+1}{3}+1=0\)
- step1: Multiply both sides of the equation by LCD:
\(\left(\frac{2x-1}{4}-\frac{x+1}{3}+1\right)\times 12=0\times 12\)
- step2: Simplify the equation:
\(2x+5=0\)
- step3: Move the constant to the right side:
\(2x=0-5\)
- step4: Remove 0:
\(2x=-5\)
- step5: Divide both sides:
\(\frac{2x}{2}=\frac{-5}{2}\)
- step6: Divide the numbers:
\(x=-\frac{5}{2}\)
The solutions to the given equations are:
(a) \( x = 24 \)
(b) \( x = \frac{3}{7} \) or \( x = \frac{3}{5} \)
(c) \( x = -\frac{5}{2} \) or \( x = -2.5 \)
Quick Answer
(a) \( x = 24 \)
(b) \( x = \frac{3}{7} \) or \( x = \frac{3}{5} \)
(c) \( x = -\frac{5}{2} \)
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