Question

question
solve the following inequality for z. write your answer in simplest form.
-6 - 2(-7z - 7) ≤ z - 1 - 10z
answer attempt 1 out of 5
z
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Explanation:

Step1: Distribute the -2

First, we distribute the -2 in \(-2(-7z - 7)\). Using the distributive property \(a(b + c)=ab+ac\), we get \(-2(-7z)-2(-7)=14z + 14\). So the left - hand side of the inequality becomes \(-6+14z + 14\). Combining like terms \(-6 + 14=8\), so the left - hand side is \(14z+8\). The inequality now is \(14z + 8\leq z-1 - 10z\).

Step2: Combine like terms on the right

On the right - hand side, combine the \(z\) terms: \(z-10z=-9z\). So the inequality is \(14z + 8\leq-9z-1\).

Step3: Add 9z to both sides

To get all the \(z\) terms on one side, we add \(9z\) to both sides of the inequality. \(14z+9z + 8\leq-9z + 9z-1\). Combining like terms, \(23z+8\leq - 1\).

Step4: Subtract 8 from both sides

Subtract 8 from both sides: \(23z+8 - 8\leq-1 - 8\). This simplifies to \(23z\leq-9\).

Step5: Divide by 23

Divide both sides of the inequality by 23 (since 23 is positive, the direction of the inequality sign remains the same). \(\frac{23z}{23}\leq\frac{-9}{23}\), so \(z\leq-\frac{9}{23}\).

Answer:

\(z\leq-\frac{9}{23}\)