Question

notation
c) \\( 7x - 2(x - 4) \leq -2 \\)
\td) \\( 6x + 1 < 9 - 2x \\)

Explanation:

Part (c)

Step 1: Distribute the -2

We use the distributive property \(a(b - c)=ab - ac\) to expand \(-2(x - 4)\).
\(7x-2(x - 4)=7x-2x + 8\)
So the inequality becomes \(7x-2x + 8\leq - 2\)

Step 2: Combine like terms

Combine the \(x\) terms: \(7x-2x = 5x\)
The inequality is now \(5x+8\leq - 2\)

Step 3: Subtract 8 from both sides

Subtract 8 from each side to isolate the term with \(x\):
\(5x+8 - 8\leq - 2-8\)
\(5x\leq - 10\)

Step 4: Divide by 5

Divide both sides by 5:
\(\frac{5x}{5}\leq\frac{-10}{5}\)
\(x\leq - 2\)

Part (d)

Step 1: Add 2x to both sides

Add \(2x\) to each side to get all \(x\) terms on one side:
\(6x + 1+2x<9-2x + 2x\)
\(8x + 1<9\)

Step 2: Subtract 1 from both sides

Subtract 1 from each side:
\(8x+1 - 1<9 - 1\)
\(8x<8\)

Step 3: Divide by 8

Divide both sides by 8:
\(\frac{8x}{8}<\frac{8}{8}\)
\(x < 1\)

Answer:

For part (c): \(x\leq - 2\)

For part (d): \(x < 1\)