Solve each inequality. Give the solution set in interval notation. 1≤(4x-5)/2<9

Verified step by step guidance

1. Start by understanding that the inequality \(1 \leq \frac{4x - 5}{2} < 9\) is a compound inequality, which means you can split it into two separate inequalities to solve simultaneously.
2. Multiply all parts of the inequality by 2 to eliminate the denominator, giving: \(2 \leq 4x - 5 < 18\).
3. Next, add 5 to all parts of the inequality to isolate the term with \(x\): \(2 + 5 \leq 4x - 5 + 5 < 18 + 5\), which simplifies to \(7 \leq 4x < 23\).
4. Now, divide all parts of the inequality by 4 to solve for \(x\): \(\frac{7}{4} \leq x < \frac{23}{4}\).
5. Finally, express the solution set in interval notation as \(\left[ \frac{7}{4}, \frac{23}{4} \right)\), where the square bracket means \(x\) can equal \(\frac{7}{4}\) and the parenthesis means \(x\) is strictly less than \(\frac{23}{4}\).

Key Concepts

Compound Inequalities

Compound inequalities involve two inequalities joined together, such as 1 ≤ (4x - 5)/2 < 9. Solving them requires finding all values of the variable that satisfy both inequalities simultaneously, often by splitting the compound inequality into two separate inequalities and solving each.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division. When multiplying or dividing by a negative number, the inequality sign must be reversed to maintain a true statement.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that an endpoint is not included, while brackets mean the endpoint is included. For example, [a, b) includes a but excludes b.