Problem 52 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it in interval notation. $$ 7 x-9<5 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, 2)\), and it's graphed as an open circle at 2 with shading to the left.

Step 1: Add 9 to Both Sides

To isolate the variable term, add 9 to both sides of the inequality. This gives us: \[ 7x - 9 + 9 < 5 + 9 \]which simplifies to\[ 7x < 14 \].

Step 2: Divide by 7

Next, divide both sides by 7 to solve for \( x \). This step gives:\[ \frac{7x}{7} < \frac{14}{7} \]which simplifies to\[ x < 2 \].

Step 3: Write in Interval Notation

The inequality \( x < 2 \) can be expressed in interval notation as \((-\infty, 2)\).This shows all values less than 2 as part of the solution set.

Step 4: Graph the Solution Set

To graph the solution set, draw a number line. Place an open circle at 2 to indicate that 2 is not included in the solution set, and shade the line to the left of 2 to show all possible values less than 2 are included. The open circle visually represents a strict inequality.

Key Concepts

Solving Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other but is rather less than, greater than, equal to and less than, or equal to and greater than the other side. Solving inequalities follows similar steps to solving equations, but there are key differences to remember:

• When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
• In our example, the inequality to solve is: \(7x - 9 < 5\).
• The goal is to isolate \(x\) on one side.
• We start by adding 9 to both sides: \(7x < 14\).
• Next, divide both sides by 7 to solve for \(x\): \(x < 2\).

By following these steps, you have successfully solved the inequality!

Graphing Inequalities

Graphing inequalities is a visual way to represent all possible solutions on a number line. For the inequality \(x < 2\), we take the following steps:

• Draw a horizontal line representing the number line.
• Locate and mark the number 2 on this line.
• Place an open circle on 2. An open circle means that the number 2 itself is not part of the solution (as the inequality is not \( \leq \)).
• Shade the region to the left of the open circle. This shaded area shows all numbers less than 2, which are solutions to the inequality \(x < 2\).

This way, you visually capture all values that satisfy the inequality.

Interval Notation

Interval notation is a method of writing down a set of numbers. It simplifies the representation of multiple numbers that complete the solution set of an inequality on a number line:

• Start with the smallest number in your solution set.
• End with the largest, using a parenthesis \(()\) to signify that the number is not included in the set (as with open circles), or a bracket \([]\) if it is included.

For the inequality \(x < 2\), we represent it in interval notation as \(( -\infty, 2 )\).

• The \( -\infty \) denotes that the values go on indefinitely towards negative infinity.
• The parenthesis around 2 shows that 2 itself isn't included in the solution.

Interval notation not only clearly describes the range but also visually complements the graph of the inequality.