Problem 22 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ 3 x>-9 $$

Step-by-Step Solution

Answer

Solution: \(x > -3\). Graph: Open circle at -3, shade right.

Step 1: Isolate the Variable

To start, you need to isolate the variable \(x\) on one side of the inequality. The inequality given is \(3x > -9\). Divide each side of the inequality by 3 to solve for \(x\). This operation will not change the direction of the inequality because we are dividing by a positive number.

Step 2: Simplify the Inequality

Perform the division: \(\frac{3x}{3} > \frac{-9}{3}\). This simplifies to \(x > -3\). This is the solution to the inequality.

Step 3: Graph the Solution Set

The solution \(x > -3\) means all values greater than -3. On a number line, you would draw a circle at -3 to indicate that -3 is not included (open circle) and shade everything to the right to show that all greater numbers are part of the solution.

Key Concepts

Solving Inequalities

Inequalities allow us to express a range of solutions rather than just a single value. When solving inequalities like the given example, the goal is to determine the range of values that satisfy the inequality condition. Here, the inequality is \(3x > -9\). To solve for \(x\), we isolate it on one side of the inequality. We accomplish this by performing the same mathematical operation on both sides. In this case, we divide both sides by 3 to solve for \(x\).
This operation gives us \(x > -3\).

• Always apply the same operation to each side of the inequality.
• When multiplying or dividing by a negative number, the inequality sign must be flipped. However, in this instance, we are dividing by a positive number so the inequality direction remains the same.

Understanding these principles empowers you to solve a wide variety of inequalities confidently.

Graphing Inequalities

Once you have solved an inequality, the next step is to graph the solution. This helps us visually understand the range of the solution. For the inequality \(x > -3\), graphing involves a number line.

• First, identify the critical value, which in this case is -3.
• On the number line, place an open circle at -3. The open circle signifies that -3 is not part of the solution since the inequality is strictly greater than (-3).
• Shade the portion of the number line to the right of -3. This shading represents all numbers greater than -3 being part of the solution.

Remember, the type of circle (open or closed) and the direction of shading depend on whether the inequality is strict (> or <) or includes equality (≥ or ≤).

Algebraic Expressions

An inequality is often expressed using an algebraic expression. Algebraic expressions are combinations of numbers, variables, and operations. The expression \(3x > -9\) involves the variable \(x\) being multiplied by 3.

• Understand that the expression represents a range rather than fixed points.
• Variables like \(x\) are placeholders for numbers that satisfy the expression.
• Operations such as addition, subtraction, multiplication, and division are used to manipulate and solve these expressions.
• The constants (in this case, 3 and -9) determine how we solve the inequality.

Mastering algebraic expressions involves recognizing how these components interact within an inequality to refine and highlight the solution set clearly. By doing so, you can both understand and analyze real-world situations described mathematically through inequalities.