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Solve for and write your answer in simplest form.Answer:
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Q. Solve for and write your answer in simplest form.Answer:
- Simplify left side: Write down the equation and simplify the left side by multiplying the fraction.The equation is (\frac{1}{2})((\frac{3}{2})x + 1) = -(-x - 1) - 3x\.\(\newlineFirst, we simplify the left side by distributing \$(\frac{1}{2})\) to both \((\frac{3}{2})x\) and \(1\).\(\newline\)\((\frac{1}{2}) \times (\frac{3}{2})x = (\frac{3}{4})x\) and \((\frac{1}{2}) \times 1 = \frac{1}{2}\).\(\newline\)So, the left side of the equation becomes \((\frac{3}{4})x + \frac{1}{2}\).
- Simplify right side: Simplify the right side of the equation.\(\newline\)The right side of the equation is \(-(-x - 1) - 3x\).\(\newline\)First, we distribute the negative sign to both \(-x\) and \(-1\), which gives us \(x + 1 - 3x\).\(\newline\)Combining like terms, we get \(-2x + 1\).\(\newline\)So, the right side of the equation simplifies to \(-2x + 1\).
- Set equal: Set the simplified left side equal to the simplified right side.\(\newline\)Now we have \((\frac{3}{4})x + \frac{1}{2} = -2x + 1\).
- Move terms: Move all terms involving \(x\) to one side of the equation and constant terms to the other side.\(\newline\)To do this, we will subtract \((3/4)x\) from both sides and subtract \(1\) from both sides.\(\newline\)This gives us \(\frac{1}{2} - 1 = -2x - \left(\frac{3}{4}\right)x\).
- Combine and simplify: Combine like terms and simplify both sides.\(\newline\)\(\frac{1}{2} - 1\) is equal to \(-\frac{1}{2}\).\(\newline\)To combine \(-2x - \frac{3}{4}x\), we need a common denominator, which is \(4\).\(\newline\)So, \(-2x\) becomes \(-\frac{8}{4}x\), and we have \(-\frac{8}{4}x - \frac{3}{4}x\).\(\newline\)Combining these gives us \(-\frac{11}{4}x\).\(\newline\)Now we have \(-\frac{1}{2} = -\frac{11}{4}x\).
- Solve for x: Solve for x by dividing both sides by the coefficient of x.\(\newline\)To isolate x, we divide both sides by \(-\frac{11}{4}\).\(\newline\)We get \(\left(-\frac{1}{2}\right) / \left(-\frac{11}{4}\right) = x\).
- Final solution: Simplify the fraction by multiplying by the reciprocal of \(-\frac{11}{4}\). Multiplying by the reciprocal, we get \((-\frac{1}{2}) \times (-\frac{4}{11}) = x\). This simplifies to \(\frac{4}{22}\), which can be reduced to \(\frac{2}{11}\). So, \(x = \frac{2}{11}\).
