Problem 30 - College Algebra

Question

Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. $$ 3 y<5-2 y<7+y $$

Step-by-Step Solution

Answer

The solution is \(-\frac{2}{3} < y < 1\) or \((-\frac{2}{3}, 1)\) in interval notation.

Step 1: Break down the compound inequality

The given compound inequality is \(3y < 5 - 2y < 7 + y\). This means we have two inequalities to solve: \(3y < 5 - 2y\) and \(5 - 2y < 7 + y\). We'll solve them one at a time.

Step 2: Solve the first inequality

Consider the first part of the inequality: \(3y < 5 - 2y\). 1. Add \(2y\) to both sides to get: \(3y + 2y < 5\) which simplifies to \(5y < 5\).2. Divide both sides by 5 to solve for \(y\): \(y < 1\).

Step 3: Solve the second inequality

Now, consider the second part: \(5 - 2y < 7 + y\).1. Subtract 5 from both sides to get: \(-2y < 2 + y\). 2. Subtract \(y\) from both sides: \(-2y - y < 2\) which simplifies to \(-3y < 2\).3. Divide both sides by -3, and remember to flip the inequality sign: \(y > -\frac{2}{3}\).

Step 4: Combine the solutions

We have the solutions \(y < 1\) and \(y > -\frac{2}{3}\). Combining these tells us that:\(-\frac{2}{3} < y < 1\).

Step 5: Express the solution in interval notation

In interval notation, the solution \(-\frac{2}{3} < y < 1\) is written as:\((-\frac{2}{3}, 1)\).

Key Concepts

Inequality Signs

Inequality signs are essential symbols in mathematics that help to express a relationship where one quantity is not necessarily equal to another. There are four commonly used inequality signs:

• The less than sign: "<"
• The greater than sign: ">"
• The less than or equal to sign: "\( \leq \)"
• The greater than or equal to sign: "\( \geq \)"

In compound inequalities, two or more inequalities are presented together to express a range of values that a variable can take. This means you use inequality signs to connect these expressions, creating a more complex condition to solve.
It's important when solving inequalities to pay attention to the direction of the inequality signs, especially when performing operations like multiplication or division by negative numbers, because these operations require you to flip the inequality sign. For example, when dividing each side of an inequality by a negative number, "\(>\)" turns to "\(<\)" and vice versa.

Interval Notation

Interval notation offers a concise way of expressing the set of solutions to inequalities. Instead of writing out inequalities, interval notation uses brackets and parentheses to show which numbers are included in the solution set.

• Parentheses "( )" indicate that the endpoint is not included in the interval (open interval).
• Brackets "[ ]" signify that the endpoint is included in the interval (closed interval).

For example, solving the compound inequality \(-\frac{2}{3} < y < 1\), the solution in interval notation is written as \((-rac{2}{3}, 1)\). This representation indicates that values of \( y \) between (but not including) \(-\frac{2}{3}\) and 1 satisfy the inequality.
Interval notation is a convenient shorthand that avoids repeating inequality signs and provides a clear range of values in a single expression.

Algebraic Solutions

Solving compound inequalities algebraically involves systematically breaking them down into simpler parts that can each be solved using basic algebraic techniques. The goal is to isolate the variable in each part of the inequality to find its possible values.

• Start by examining the compound inequality and splitting it into two or more separate inequalities.
• Use algebraic operations like addition, subtraction, multiplication, or division to manipulate each inequality, isolating the variable.
• Keep track of the changes you make, especially when multiplying or dividing by negative numbers because the inequality sign must be reversed.

In our example, the compound inequality \(3y < 5 - 2y < 7 + y\) is split into two components: \(3y < 5 - 2y\) and \(5 - 2y < 7 + y\). Solving these individually gives values for \(y\) that are then combined to form a final solution.
This step-by-step approach ensures that the entire range of possible solutions is accounted for and logically connected, allowing us to express the solution both in terms of inequality and interval notation.