Question

linear inequalities with decimals
question
solve the following inequality:
$-7 \leq -3(5x - 7.6) + 2$
answer

Explanation:

Step1: Simplify the right - hand side

First, we need to simplify the expression \(-3(5x - 7.6)+2\). Using the distributive property \(a(b - c)=ab - ac\), where \(a=-3\), \(b = 5x\) and \(c = 7.6\), we get:
\(-3\times5x-(-3)\times7.6 + 2=-15x + 22.8+2\)
Then, combine like terms: \(-15x+(22.8 + 2)=-15x+24.8\)
So the inequality becomes \(-7\leq - 15x+24.8\)

Step2: Subtract 24.8 from both sides

Subtract \(24.8\) from both sides of the inequality \(-7\leq - 15x+24.8\) to isolate the term with \(x\).
\(-7-24.8\leq - 15x+24.8 - 24.8\)
Calculate \(-7-24.8=-31.8\), and the right - hand side simplifies to \(-15x\). So we have \(-31.8\leq - 15x\)

Step3: Divide both sides by - 15 (and reverse the inequality sign)

When we divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
Divide both sides of \(-31.8\leq - 15x\) by \(-15\):
\(\frac{-31.8}{-15}\geq\frac{-15x}{-15}\)
Calculate \(\frac{-31.8}{-15}=2.12\) and \(\frac{-15x}{-15}=x\). So we get \(2.12\geq x\) or \(x\leq2.12\)

Answer:

\(x\leq2.12\)