Linear Inequalities

The set of all integer values of x that satisfy the inequality 19 ≤ 3x ≤ 27 is {7, 8, 9}.

Explanation:

To solve the inequality 19 ≤ 3x ≤ 27, we first divide each part of the inequality by 3:

19 ÷ 3 ≤ x ≤ 27 ÷ 3

Which simplifies to:

6.33 ≤ x ≤ 9

The integer values of x that satisfy this inequality are the integers between 6.33 and 9, inclusive, which are {7, 8, 9}.

\noindent \textbf{Solution:} \\ We are given the inequality , and the solution set is . The inequality can be rewritten as two separate inequalities: Now, solve for by isolating it in terms of . To find the endpoints of the solution set, substitute and into the inequality. 1. When : Adding 8 to all sides: 2. When : Subtracting 4 from all sides: By combining the two conditions, we find that the value of is , which is the midpoint of the intersection of the two ranges.

The correct option is (D) : -4

We are given the inequality:

Step 1: Solve the inequality

We need to solve the compound inequality .

First part:

Multiply both sides of the inequality by 2 to eliminate the denominator: Now subtract 2 from both sides: Finally, divide both sides by 3:

Second part:

Multiply both sides of the inequality by 2 to eliminate the denominator: Now subtract 2 from both sides: Finally, divide both sides by 3:

Step 2: Combine the results

From the first part, we found that , and from the second part, we found that . Thus, the solution set is:

The correct option is (C) :