Question

the following inequality and graph the solution. 3(x + 5) - 6 > -2(x - 2) + 15
choose the correct graph.
help me solve this
view an example
get more help

Explanation:

Step1: Expand both sides

First, expand the left - hand side \(3(x + 5)-6\) using the distributive property \(a(b + c)=ab+ac\). We get \(3x+15 - 6=3x + 9\).
Then, expand the right - hand side \(-2(x - 2)+15\). Using the distributive property, we have \(-2x + 4+15=-2x+19\).
So the inequality becomes \(3x + 9>-2x + 19\).

Step2: Add \(2x\) to both sides

To get all the \(x\) terms on one side, we add \(2x\) to both sides of the inequality \(3x + 9>-2x + 19\).
\(3x+2x + 9>-2x+2x + 19\), which simplifies to \(5x+9>19\).

Step3: Subtract 9 from both sides

Subtract 9 from both sides of the inequality \(5x + 9>19\).
\(5x+9 - 9>19 - 9\), which gives \(5x>10\).

Step4: Divide both sides by 5

Divide both sides of the inequality \(5x>10\) by 5.
\(\frac{5x}{5}>\frac{10}{5}\), so \(x > 2\).

The graph of the solution \(x>2\) should have an open circle at \(x = 2\) (since \(x\) is not equal to 2) and an arrow pointing to the right (since \(x\) is greater than 2). Looking at the given graphs, the graph that represents \(x>2\) is the one with an open circle at 2 and the arrow going to the right (assuming the correct graph among A, B, C, D is the one with this feature, for example, if OB is the graph with open circle at 2 and arrow right, then the answer is OB).

Answer:

The solution of the inequality is \(x > 2\), and the correct graph is the one (e.g., OB) that shows an open circle at \(x = 2\) and the arrow pointing to the right.