Question

solve the inequality and graph the solution.
2(n - 2) + 5 < 3
plot the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.

Explanation:

Step1: Expand the left - hand side

Use the distributive property \(a(b + c)=ab+ac\). Here \(a = 2\), \(b=n\), \(c=-2\). So \(2(n - 2)+5=2n-4 + 5\). Then \(2n-4 + 5=2n+1\). The inequality becomes \(2n+1\lt3\).

Step2: Isolate the variable term

Subtract \(1\) from both sides of the inequality. Using the subtraction property of inequalities \(a\lt b\) implies \(a - c\lt b - c\). So \(2n+1-1\lt3 - 1\), which simplifies to \(2n\lt2\).

Step3: Solve for \(n\)

Divide both sides of the inequality by \(2\). Using the division property of inequalities \(a\lt b\) and \(c>0\) implies \(\frac{a}{c}\lt\frac{b}{c}\). So \(\frac{2n}{2}\lt\frac{2}{2}\), which gives \(n\lt1\).

Answer:

The solution of the inequality \(2(n - 2)+5\lt3\) is \(n\lt1\). On the number - line, we have an open circle at \(n = 1\) (because the inequality is strict, i.e., \(n
eq1\)) and the line extends to the left of \(n = 1\) towards \(-\infty\).