Question

solve the inequality. graph the solution set and write it in interval notation. 4x^2 + 6x - 10 < 4x(x - 5) (use integers or fractions for any numbers in the expression.) choose the correct graph below. write the answer in interval notation.

Explanation:

Step1: Expand the right - hand side

First, expand \(4x(x - 5)=4x^{2}-20x\). The inequality \(4x^{2}+6x - 10<4x(x - 5)\) becomes \(4x^{2}+6x - 10<4x^{2}-20x\).

Step2: Subtract \(4x^{2}\) from both sides

Subtracting \(4x^{2}\) from both sides of the inequality \(4x^{2}+6x - 10<4x^{2}-20x\), we get \(6x-10<-20x\).

Step3: Add \(20x\) to both sides

Adding \(20x\) to both sides gives \(6x + 20x-10<0\), which simplifies to \(26x-10<0\).

Step4: Add 10 to both sides

Adding 10 to both sides, we have \(26x<10\).

Step5: Solve for \(x\)

Dividing both sides by 26, we get \(x<\frac{10}{26}=\frac{5}{13}\).

The solution in interval notation is \((-\infty,\frac{5}{13})\).

To graph this, we have an open - circle at \(x = \frac{5}{13}\) and an arrow pointing to the left.

Answer:

The solution in interval notation is \((-\infty,\frac{5}{13})\). The correct graph is one with an open - circle at \(x=\frac{5}{13}\) and an arrow pointing to the left (depending on the options, it would be the graph that represents all real numbers less than \(\frac{5}{13}\)).