Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(2|x+10| \geq 9\)

Step-by-Step Solution

Answer

The solution of the inequality \(2|x+10| \geq 9\) is \(x \leq -14.5 \) and \(x \geq -5.5 \). On a number line, this is represented by two separate shaded regions, one to the left of -14.5 and one to the right of -5.5.

Step 1: Rewrite the Absolute Value Inequality as Compound Inequality

First, isolate the absolute value expression. Divide both sides of the inequality by 2 to yield \(|x+10| \geq 4.5\). Now, express this as a compound inequality. It means \(x+10 \geq 4.5\) or \(x+10 \leq -4.5\) based on the definition of absolute value which is always positive or zero.

Step 2: Solve each Inequality

Now, solve each inequality separately. For \(x+10 \geq 4.5\), subtract 10 from both sides to get \(x \geq -5.5\). For \(x+10 \leq -4.5\), subtract 10 from both sides to get \(x \leq -14.5\).

Step 3: Graph the Solution

Now, graph the solution set on a number line. Draw a closed circle at -14.5 and shade to the left for \(x \leq -14.5\). Then, draw a closed circle at -5.5 and shade to the right for \(x \geq -5.5\). Since the inequality does not include the values in between -14.5 and -5.5, the shaded regions will not overlap.

Key Concepts

Compound Inequalities

Understanding compound inequalities is crucial to solving problems involving ranges of possible solutions. A compound inequality consists of two separate inequalities joined by the word 'and' or 'or'. When the word 'and' is used, both conditions described by the inequalities must be true simultaneously. Conversely, 'or' indicates that either of the conditions being true forms a valid solution. For instance, the inequality 'a < x < b' represents 'x' is greater than 'a' AND 'x' is less than 'b', which means 'x' must lie between 'a' and 'b'.

In the context of absolute value inequalities, such as the exercise \(2|x+10| \geq 9\), translating the inequality into a compound inequality helps to consider the range of values that satisfy the original condition. The expression \(|x+10| \geq 4.5\) is equivalent to the compound inequality \(x+10 \geq 4.5\) OR \(x+10 \leq -4.5\), accounting for the nature of absolute values producing both positive and negative results. It's essential to remember that 'and' narrows the solution set while 'or' expands it, fitting the solutions for both parts of the absolute value expression.

Absolute Value Expression

An absolute value expression refers to the distance of a number from zero on a number line, without considering direction. It's denoted by two vertical bars on either side of the number or expression, such as \( |x| \). The absolute value of a number is always non-negative, meaning it's either positive or zero. The absolute value of \( x \) essentially asks: How far is \( x \) from zero?

When working with absolute values in inequalities, like \(2|x+10| \geq 9\), the expression can have two possible outcomes—since a distance is positive by definition, the inside of the absolute value can be either positive or negative without affecting the distance it represents. Hence, solving \( |x+10| \geq 4.5 \) requires evaluating both \( x+10 \geq 4.5 \) (when \( x+10 \) is already positive or zero) and \( x+10 \leq -4.5 \) (which is the equivalent distance in the negative direction). This is why it is transformed into a compound inequality in the context of the exercise.

Number Line Graphing

Number line graphing is a visual representation of numbers or a range of numbers on a straight horizontal line where each point corresponds to a real number. In inequalities, graphing on a number line helps to visually demonstrate all the possible solutions that satisfy the inequality. The crucial components of a number line graph when used for inequalities include the use of open or closed circles and shading.

Closed circles on the number line are used when the number is part of the solution set, for example when an inequality is 'greater than or equal to' (\geq) or 'less than or equal to' (\leq). In comparison, open circles mean the number itself is not included in the solution set, as with 'greater than' (>) or 'less than' (<) scenarios.

Shading the number line represents the set of numbers that are solutions to the inequality. If the inequality uses 'and', the shading is between the two points, whereas with 'or', the line is shaded in opposite directions from the two points. In the given exercise \(2|x+10| \geq 9\), once we find the two critical values, -14.5 and -5.5, we place closed circles on them and shade to the left of -14.5 and to the right of -5.5, as the inequality includes those numbers (represented with closed circles) and every number to the respective sides.