Question

linear inequalities with decimals
question
solve the following inequality:
$-8x \geq -8x - 6.3(-4x + 0.6)$

Explanation:

Step1: Expand the right - hand side

First, we need to expand the term \(-6.3(-4x + 0.6)\) using the distributive property \(a(b + c)=ab+ac\). Here, \(a=-6.3\), \(b = - 4x\) and \(c = 0.6\). So we have:
\(-6.3\times(-4x)+(-6.3)\times0.6=25.2x-3.78\)
The original inequality \(-8x\geq - 8x-6.3(-4x + 0.6)\) becomes:
\(-8x\geq - 8x + 25.2x-3.78\)

Step2: Simplify the right - hand side

Combine like terms on the right - hand side. The \(-8x\) and \(25.2x\) are like terms. So \(-8x + 25.2x=17.2x\). The inequality is now:
\(-8x\geq17.2x - 3.78\)

Step3: Add \(8x\) to both sides

To get all the \(x\) terms on one side, we add \(8x\) to both sides of the inequality.
\(-8x + 8x\geq17.2x+8x - 3.78\)
Simplifying the left - hand side (\(-8x + 8x = 0\)) and the right - hand side (\(17.2x+8x=25.2x\)), we get:
\(0\geq25.2x - 3.78\)

Step4: Add 3.78 to both sides

Add \(3.78\) to both sides of the inequality:
\(0 + 3.78\geq25.2x-3.78 + 3.78\)
Simplifying, we have:
\(3.78\geq25.2x\)

Step5: Divide both sides by 25.2

Now, divide both sides of the inequality by \(25.2\) to solve for \(x\). Since \(25.2>0\), the direction of the inequality sign remains the same.
\(\frac{3.78}{25.2}\geq x\)
Calculating \(\frac{3.78}{25.2}\), we can multiply the numerator and denominator by 100 to get \(\frac{378}{2520}\), and then divide numerator and denominator by 126: \(\frac{378\div126}{2520\div126}=\frac{3}{20} = 0.15\). So \(x\leq0.15\)

Answer:

\(x\leq0.15\)