Problem 76 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ 2-2[3 h-(7-h)]>6[-(19+h)-(1-h)] $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 17)\).

Step 1: Simplify Inside the Brackets

Start by simplifying the expressions inside the brackets. For the left side, simplify \[ 3h - (7 - h) = 3h - 7 + h = 4h - 7. \]For the right side, simplify \[-(19 + h) - (1 - h) = -19 - h - 1 + h = -20. \] Now rewrite the inequality as:\[ 2 - 2(4h - 7) > 6(-20). \]

Step 2: Distribute the Terms

Distribute the constants into the terms within the parentheses. On the left side:\[ 2 - 2(4h - 7) = 2 - 8h + 14 = 16 - 8h. \]On the right side:\[ 6(-20) = -120. \]Thus, the inequality becomes:\[ 16 - 8h > -120. \]

Step 3: Solve for h

Isolate the term with \( h \). Subtract 16 from both sides of the inequality:\[ 16 - 8h - 16 > -120 - 16, \]which simplifies to:\[ -8h > -136. \]Now divide both sides by -8, remembering to reverse the inequality sign when dividing by a negative:\[ h < \frac{136}{8} = 17. \]

Step 4: Write in Interval Notation

The inequality \( h < 17 \) is all numbers less than 17. In interval notation, this is expressed as:\[ (-\infty, 17). \]

Step 5: Graph the Solution

On a number line, represent the solution by shading all of the numbers to the left of 17 and placing an open circle at 17, indicating 17 is not included in the solution set.

Key Concepts

Interval Notation

Interval notation is a shorthand used in mathematics to describe a set of numbers between two endpoints. Instead of listing every number in the set, interval notation provides a concise way to represent ranges. It uses parentheses

• "(" or ")" indicate endpoints that are excluded from the set.
• "[" or "]" depict endpoints that are included in the set.

In the inequality solving from our original exercise, we arrived at a solution where \( h < 17 \). This means all numbers less than 17 but not including 17 itself are part of our solution. In interval notation, this is written as

• \((-\infty, 17)\).

The use of \((-\infty\)) implies that our solution extends indefinitely to the left. Parentheses are used around 17, highlighting that it is not part of our solution set, but the set extends up close to 17.

Graphing Inequalities

Graphing inequalities is a visual method for showing all possible solutions of an inequality. Drawing these solutions on a number line helps with understanding the relationship between numbers and the inequality symbol.
In our example, we need to graph \( h < 17 \). Here's how:

• Draw a horizontal line to represent the number line.
• Mark a point at 17.
• Place an open circle at 17. This open circle shows that 17 is not included in the solution set.
• Shade the entire section of the number line left of the 17 since all numbers smaller than 17 are included in the solution.

This method of graphing makes it easy to visually interpret the information contained in the inequality and understand which numbers satisfy the condition.

Distributive Property

The distributive property is a key algebraic principle used when simplifying expressions, especially those involving parentheses. It states that multiplying a single term by terms within a parenthesis can be done by distributing the single term to each term inside. This is expressed as

• \( a(b + c) = ab + ac \).

In our original inequality, on the left-hand side expression \( 2 - 2(4h - 7) \):

• We used the distributive property to simplify \(-2(4h - 7)\).
• This means \(-2 ext{times } 4h\) becomes \(-8h\) and \(-2 ext{ times } -7\) gives \(+14\).

Thus, the expression simplifies to \( 16 - 8h \) once the entire left-hand side is addressed. Understanding how to correctly apply the distributive property is essential not only for solving inequalities but for simplifying many algebraic expressions.