Problem 1 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(a+2<3.5\)

Step-by-Step Solution

Answer

The solution is \(a < 1.5\); on a number line, use an open circle at 1.5 and shade left.

Step 1: Isolate the Variable

Begin by isolating the variable \(a\) in the inequality \(a + 2 < 3.5\). To do this, subtract 2 from both sides of the inequality: \(a + 2 - 2 < 3.5 - 2\). This simplifies to \(a < 1.5\).

Step 2: Graph the Solution on a Number Line

To graph \(a < 1.5\) on a number line, draw a number line and place an open circle at 1.5. Then, shade the line to the left of 1.5 to indicate all numbers less than 1.5 are included in the solution set.

Key Concepts

Solving Inequalities

Solving inequalities involves finding the set of values that make an inequality true. An inequality is expressed using symbols such as <, >, ≤, or ≥. They are like equations, but instead of showing equality between two expressions, they show that one side is greater or less than the other.

When solving the inequality \(a + 2 < 3.5\), our main goal is to isolate the variable \(a\). This process is similar to solving equations. You perform operations to move terms around while respecting the inequality.

• Subtract or add the same value to both sides.
• Multiply or divide both sides by the same number. Caution: if you multiply or divide by a negative number, you must flip the inequality sign.

In the example \(a + 2 < 3.5\), subtracting 2 from both sides simplifies it to \(a < 1.5\). This means \(a\) can be any number less than 1.5.

Graphing Inequalities

Graphing inequalities is a visual way to represent the solution set of an inequality. It helps you see all possible solutions and how they relate to each other.

To graph \(a < 1.5\), we use a number line. The solution tells us that \(a\) can take any value less than 1.5. Therefore, we start by marking 1.5 on the number line. To express that 1.5 itself is not part of the solution, we draw an open circle (as opposed to a closed circle, which would mean that number is included).

Finally, shade the portion of the number line that includes all numbers less than 1.5. The line represents all potential values of \(a\). This visual representation is helpful in understanding the scope of possible solutions for \(a\).

Number Line Representation

A number line is a straightforward tool for representing numbers, including solutions to inequalities. It helps illustrate relationships between numbers and the extent of their values in a clear and simple manner.

When you create a number line representation for \(a < 1.5\), start by sketching a horizontal line. Mark points on it to indicate significant numbers, like whole numbers or fractions, including where 1.5 sits. Place an open circle at 1.5 to show it is not included in our solution set, and shade everything to the left of this point.

• Open circles show numbers not included (like in \(a < 1.5\)).
• Closed circles show numbers that are included (like in \(a \leq 1.5\)).

This number line representation serves as a quick reference to verify or communicate the solutions. It's also useful for comparing multiple inequalities or understanding how different conditions interact on the number line.