Question

solve the linear inequality for a. graph your answer in the number line. then write the solution in inequality notation. \\(-11 \geq a - \frac{17}{2}\\) show your work here hint: to add inequalities (<, >, ≤, ≥), type \less\ or \greater\ (number line and toggle buttons here) solution: solve the linear inequality for n. graph your answer in the number line. then write the solution in inequality notation. \\(-21 > -4n - 9\\) show your work here hint: to add inequalities (<, >, ≤, ≥), type \less\ or \greater\ (number line here)

Explanation:

Solving for \( a \):

Step1: Add \( \frac{17}{2} \) to both sides

To isolate \( a \), we add \( \frac{17}{2} \) to both sides of the inequality \( -11 \geq a - \frac{17}{2} \).
\( -11 + \frac{17}{2} \geq a \)
First, convert \( -11 \) to a fraction with denominator 2: \( -11 = -\frac{22}{2} \)
So, \( -\frac{22}{2} + \frac{17}{2} = -\frac{5}{2} = -2.5 \)
Thus, \( -2.5 \geq a \) or \( a \leq -\frac{5}{2} \)

Step2: Graph on number line

For the number line, we have a closed circle at \( -2.5 \) (since the inequality is \( \leq \)) and the line shaded to the left (towards more negative numbers).

Step3: Inequality notation

The solution in inequality notation is \( a \leq -\frac{5}{2} \) (or \( a \leq -2.5 \))

Solving for \( n \):

Step1: Add 9 to both sides

Start with \( -21 > -4n - 9 \). Add 9 to both sides:
\( -21 + 9 > -4n \)
\( -12 > -4n \)

Step2: Divide by -4 (reverse inequality)

When dividing an inequality by a negative number, we reverse the inequality sign. Divide both sides by \( -4 \):
\( \frac{-12}{-4} < n \)
\( 3 < n \) or \( n > 3 \)

Step3: Graph on number line

For the number line, we have an open circle at 3 (since the inequality is \( > \), not \( \geq \)) and the line shaded to the right (towards larger numbers).

Step4: Inequality notation

The solution in inequality notation is \( n > 3 \)

Answer:

(for \( a \)):
\( a \leq -\frac{5}{2} \) (or \( a \leq -2.5 \))