Question

a) -3(x + 5) > 6
-3x - 5 > 6
c) x - 2x ≥ -5

Explanation:

Step1: Solve for part (a)

We have the inequality \(-3(x + 5)>6\). First, we divide both sides by \(-3\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So, dividing both sides by \(-3\) gives:
\(x + 5<\frac{6}{-3}\)
\(x + 5<-2\)
Then, we subtract \(5\) from both sides:
\(x<-2 - 5\)
\(x<-7\)

Wait, there was a mistake in the original note's second line (it should be \(-3x-15>6\) when expanding \(-3(x + 5)\), not \(-3x - 5>6\)). Let's correct that.
Correct Step1 for part (a): Expand \(-3(x + 5)\)
Using the distributive property \(a(b + c)=ab+ac\), here \(a=-3\), \(b = x\), \(c = 5\), so \(-3(x + 5)=-3x-15\). So the inequality is \(-3x-15>6\).

Step2: Solve the corrected part (a) inequality

Add \(15\) to both sides:
\(-3x-15 + 15>6 + 15\)
\(-3x>21\)
Divide both sides by \(-3\) (and reverse the inequality sign):
\(x<\frac{21}{-3}\)
\(x<-7\)

Step3: Solve for part (c)

We have the inequality \(x-2x\geq - 5\). Combine like terms:
\((1 - 2)x\geq - 5\)
\(-x\geq - 5\)
Multiply both sides by \(-1\) (and reverse the inequality sign):
\(x\leq5\)

Answer:

For part (a): \(x < - 7\)
For part (c): \(x\leq5\)