Problem 15 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-2 x-2 \leq 1+x$$

Step-by-Step Solution

Answer

The solution set is \( [-1, \infty) \).

Step 1 - Move all terms involving x to one side

Start by moving the variable terms to one side of the inequality. Subtract x from both sides to get:\[ -2x - x - 2 \leq 1 \]

Step 2 - Simplify the inequality

Combine the like terms on the left side:\[ -3x - 2 \leq 1 \]

Step 3 - Isolate the term with x

Add 2 to both sides of the inequality to isolate the term with x:\[ -3x \leq 3 \]

Step 4 - Solve for x

Divide both sides by -3 and remember to reverse the inequality when dividing by a negative number:\[ x \geq -1 \]

Step 5 - Write the solution in interval notation

The solution to the inequality \( -2x-2 \leq 1+x \) in interval notation is:\[ [-1, \infty) \]

Key Concepts

inequality notation

Inequalities are mathematical expressions involving the symbols \(eq, <, \leq, >, \geq\) to show the relationship between two values. They indicate that one side of the expression is either greater than, less than, or not equal to the other. There are four main types of inequalities:

• Less than: \( < \)
• Greater than: \( > \)
• Less than or equal to: \( \leq \)
• Greater than or equal to: \( \geq \)

In the original exercise, the inequality is \(-2x-2 \leq 1+x\). To solve inequalities, it's essential to understand the notation and properties, especially when manipulating these expressions. For example, when multiplying or dividing by a negative number, we must reverse the inequality sign. This is crucial for arriving at the correct solution.

interval notation

Interval notation is a way of describing a range of values that an inequality encompasses. It uses brackets and parentheses to represent the set of solutions:

• \((a, b)\): Values between \(a\) and \(b\), not inclusive.
• \([a, b]\): Values between \(a\) and \(b\), inclusive.
• \((-\infty, b)\): All values less than \(b\).
• \((a, \infty)\): All values greater than \(a\).

For the given inequality \(-2x-2 \leq 1+x\), after solving we get \(x \geq -1\). In interval notation, this is written as \([-1, \infty)\). This notation tells us that the solution includes all values starting from \(-1\) to positive infinity, where \(-1\) is included.

solving linear inequalities

Solving linear inequalities involves a few systematic steps:

• Isolate the variable: Get all variable terms on one side of the inequality and constants on the other.
• Simplify the expression: Combine like terms to simplify.
• Manipulate the inequality: Perform necessary operations to isolate the variable, noting that multiplying or dividing by negatives reverses the inequality sign.

In the original exercise, we start with \(-2x-2 \leq 1+x\). Following the steps:
1. Move all \(x\) terms to one side: \(-2x - x - 2 \leq 1\)
2. Simplify: \(-3x - 2 \leq 1\)
3. Move constant to the other side: \(-3x \leq 3\)
4. Solve for \(x\): \(x \geq -1\)
Finally, write the solution using interval notation: \([-1, \infty)\). This systematic approach ensures accuracy while solving linear inequalities.