Problem 55 - Algebra for College Students

Question

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(-3<2-x<3\)

Step-by-Step Solution

Answer

The solution is \((-1, 5)\).

Step 1: Understand the Inequality

We have a compound inequality: \(-3 < 2 - x < 3\). This means we need to find the values of \(x\) that make both inequalities true: 1. \(-3 < 2 - x\) 2. \(2 - x < 3\).

Step 2: Solve the First Inequality

Focus on the inequality \(-3 < 2 - x\):1. Subtract 2 from both sides to isolate \(-x\): \(-3 - 2 < -x\) simplifies to \(-5 < -x\).2. Multiply both sides by \(-1\) (and remember to flip the inequality sign):\(5 > x\).

Step 3: Solve the Second Inequality

Focus on the inequality \(2 - x < 3\):1. Subtract 2 from both sides to isolate \(-x\):\(2 - 2 - x < 3 - 2\) simplifies to \(-x < 1\).2. Multiply both sides by \(-1\) (and remember to flip the inequality sign):\(x > -1\).

Step 4: Combine the Solutions

We now combine the two results from the inequalities:1. From Step 2, the solution is \(x < 5\).2. From Step 3, the solution is \(x > -1\).3. Combine: \(-1 < x < 5\).

Step 5: Express in Interval Notation

Now express the combined solution \(-1 < x < 5\) in interval notation. This corresponds to the open interval: \((-1, 5)\).

Key Concepts

Interval Notation

Interval notation is a way to describe subsets of real numbers on a number line. Instead of writing inequalities, you can succinctly express the range of values in a more compact form. For instance, the interval \((-1, 5)\) describes all numbers between -1 and 5, excluding -1 and 5 themselves. This is known as an open interval because it does not include its endpoints.

In interval notation:

• Round brackets \( ( ) \) represent open intervals, which do not include endpoints.
• Square brackets \( [ ] \) indicate closed intervals, including endpoints.
• Mixing brackets \( ( ] \) or \( [ ) \) allows for one-sided inclusion, like \( (a, b] \) which includes b but not a.

Interval notation is particularly helpful in representing solution sets of inequalities, making it clear when endpoints are included or not. This compact notation is widely used in mathematics, making it integral for understanding solution sets accurately.

Solving Inequalities

Inequalities express the relationship between two values, indicating if one value is larger or smaller than the other. Solving inequalities involves finding the set of all possible values that an unknown variable can take to make the inequality true.

In a compound inequality such as \(-3 < 2-x < 3\), you need to solve two inequalities at once:

• The first inequality is \(-3 < 2-x\).
• The second inequality is \(2-x < 3\).

To solve each, you'll generally perform similar steps as you would with equations:

• Isolate the variable on one side.
• Use inverse operations like addition, subtraction, multiplication, or division.
• Remember that multiplying or dividing by a negative number flips the inequality sign.

Successfully solving both inequalities allows you to combine the results, giving a range of values satisfying both conditions. This combination is often expressed in interval notation for clarity.

Algebraic Solution Steps

When faced with a compound inequality, the algebraic solution involves breaking down the problem into manageable parts. Each step focuses on isolating the variable to determine the values that satisfy the inequality.

For the compound inequality \(-3 < 2 - x < 3\), follow these algebraic solution steps:

• Step 1: Solve the first inequality \(-3 < 2 - x\). Subtract 2 from both sides to simplify to \(-5 < -x\). Then, multiply by -1 to get \(5 > x\).
• Step 2: Solve the second inequality \(2 - x < 3\). Again, subtract 2 from both sides to get \(-x < 1\). Multiply by -1, flipping the inequality, so \(x > -1\).
• Step 3: Combine the solutions from both steps: \(-1 < x < 5\). This shows the range where the variable x satisfies both inequalities.

Understanding these algebraic steps is essential because it not only helps in finding the correct solution, but also in consistently applying similar strategies to other inequality problems. The final solution can then be expressed using interval notation, which in this case is \((-1, 5)\).