Problem 13 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-2 x+8 \leq 16$$

Step-by-Step Solution

Answer

The solution set in interval notation is \([-4, \infty)\).

Step 1 - Isolate the variable term

First, subtract 8 from both sides of the inequality to isolate the term with the variable. \[ -2x + 8 - 8 \leq 16 - 8 \] This simplifies to: \[ -2x \leq 8 \]

Step 2 - Solve for the variable

Next, divide both sides of the inequality by -2. Remember, dividing by a negative number reverses the inequality direction: \[ x \geq \frac{8}{-2} \] This simplifies to: \[ x \geq -4 \]

Step 3 - Write the solution in interval notation

The solution \( x \geq -4 \) means that x can be any number greater than or equal to -4. In interval notation, this is written as: \[ [-4, \infty) \]

Key Concepts

Interval Notation

When we solve inequalities, writing the solution in interval notation is a clear and standardized way to represent the range of possible values for the variable. Interval notation uses brackets and parentheses to describe these ranges. Here's a quick guide:

• [a, b]: Includes both endpoints a and b (closed interval).
• (a, b]: Excludes the start point a, includes the endpoint b (half-open interval).
• [a, b): Includes the start point a, excludes the endpoint b (half-open interval).
• (a, b): Excludes both endpoints a and b (open interval).

This notation helps us see at a glance whether endpoints are included (denoted by square brackets) or excluded (denoted by round brackets). For example, the interval notation \[(-4, \infty)\] means all numbers greater than or equal to -4 extending to infinity, where the infinity part always uses a parenthesis because infinity is never a finite value.

Inequality Reversal

One essential rule to remember when solving inequalities is the effect of multiplying or dividing by a negative number. This action requires reversing the inequality sign. Here’s why: Suppose we have the inequality \[7 > 3\]. Clearly, this is true. But if we multiply both sides by -1, we get: \[-7 > -3\]. This statement is false because -7 is actually less than -3. To correct it, we must reverse the symbol: \[-7 < -3\]. Similarly, when solving \[-2x \leq 8\], we divide both sides by -2, yielding \[x \geq -4\]. Reversing the inequality ensures the truth of the relationship between the quantities remains intact. Always be vigilant about this rule when working with inequalities.

Linear Inequalities

Linear inequalities, such as \[-2x + 8 \leq 16\], involve variables raised to the first power. Solving them is somewhat like solving linear equations, but with one key difference – the possibility of reversing the inequality when multiplying or dividing by negative numbers. Here’s a structured approach to solving them:

• Isolate the variable: Separate the term with the variable from the constants. For the given equation, subtract 8 from both sides: \[-2x + 8 - 8 \leq 16 - 8\], simplifying to \[-2x \leq 8\].
• Solve for the variable: Divide or multiply to get the variable alone. In this case, divide by -2 (and remember to reverse the inequality): \[x \geq -4\].
• Express in interval notation: Translate the inequality to interval form. For \[x \geq -4\], the interval notation is \[[-4, \infty)\].

Understanding the method ensures a consistent and correct approach to tackling these problems.