Question

1. solve the inequality. graph the solution on a number line and write the final answer as an inequality. please show your work.

4(2n + 5) ≥ 3n - 5
type a response

Explanation:

Step1: Expand the left side

We start with the inequality \(4(2n + 5)\geq3n - 5\). First, we use the distributive property \(a(b + c)=ab+ac\) to expand the left - hand side. So, \(4\times2n+4\times5\geq3n - 5\), which simplifies to \(8n + 20\geq3n-5\).

Step2: Subtract \(3n\) from both sides

To get all the \(n\) terms on one side, we subtract \(3n\) from both sides of the inequality. \((8n + 20)-3n\geq(3n - 5)-3n\). This gives us \(5n+20\geq - 5\).

Step3: Subtract 20 from both sides

Next, we want to isolate the term with \(n\). We subtract 20 from both sides of the inequality: \((5n + 20)-20\geq-5 - 20\). This simplifies to \(5n\geq-25\).

Step4: Divide both sides by 5

To solve for \(n\), we divide both sides of the inequality by 5. Since 5 is a positive number, the direction of the inequality sign remains the same. \(\frac{5n}{5}\geq\frac{-25}{5}\), which gives us \(n\geq - 5\).

Answer:

The solution to the inequality \(4(2n + 5)\geq3n - 5\) is \(n\geq - 5\). To graph this on a number line, we draw a closed circle at \(-5\) (because the inequality is "greater than or equal to") and shade the region to the right of \(-5\).