Question

linear inequalities with fractions
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solve the following inequality:
\\(\frac{3}{4}x - (-5x + 3) \leq -x\\)

Explanation:

Step1: Simplify the left side

First, simplify the left - hand side of the inequality \(\frac{3}{4}x-(-5x + 3)\leq -x\).
We know that subtracting a negative is like adding a positive, so \(\frac{3}{4}x + 5x-3\leq -x\).
Combine the \(x\) - terms on the left: \(\frac{3}{4}x+5x=\frac{3}{4}x+\frac{20}{4}x=\frac{3 + 20}{4}x=\frac{23}{4}x\).
So the inequality becomes \(\frac{23}{4}x-3\leq -x\).

Step2: Add \(x\) to both sides

To get all the \(x\) - terms on one side, add \(x\) to both sides of the inequality \(\frac{23}{4}x-3\leq -x\).
\(\frac{23}{4}x+x-3\leq -x + x\).
Since \(-x + x = 0\) and \(x=\frac{4}{4}x\), then \(\frac{23}{4}x+\frac{4}{4}x-3\leq0\).
Combining the \(x\) - terms: \(\frac{23 + 4}{4}x-3\leq0\), or \(\frac{27}{4}x-3\leq0\).

Step3: Add 3 to both sides

Add 3 to both sides of the inequality \(\frac{27}{4}x-3\leq0\).
We get \(\frac{27}{4}x-3 + 3\leq0 + 3\), which simplifies to \(\frac{27}{4}x\leq3\).

Step4: Multiply both sides by \(\frac{4}{27}\)

To solve for \(x\), multiply both sides of the inequality \(\frac{27}{4}x\leq3\) by \(\frac{4}{27}\).
Since \(\frac{4}{27}\) is a positive number, the direction of the inequality sign remains the same.
\(x\leq3\times\frac{4}{27}\).
Simplify the right - hand side: \(3\times\frac{4}{27}=\frac{12}{27}=\frac{4}{9}\).

Answer:

\(x\leq\frac{4}{9}\)