Problem 20 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ 7 x+3<9 x-3 x $$

Step-by-Step Solution

Answer

Set notation: \( \{ x \mid x < -3 \} \). Graph: open circle on \(-3\), shade left.

Step 1: Simplify the Given Inequality

The given inequality is \( 7x + 3 < 9x - 3x \). First, simplify the right side of the inequality. The expression \( 9x - 3x \) simplifies to \( 6x \). So, the inequality becomes \( 7x + 3 < 6x \).

Step 2: Isolate x on One Side

To solve for \( x \), subtract \( 6x \) from both sides of the inequality: \( 7x + 3 - 6x < 6x - 6x \), which simplifies to \( x + 3 < 0 \).

Step 3: Solve for x

Subtract 3 from both sides of the inequality to isolate \( x \): \( x + 3 - 3 < 0 - 3 \), resulting in \( x < -3 \).

Step 4: Write Solution in Set Notation

The solution set in set notation is \( \{ x \mid x < -3 \} \).

Step 5: Graph the Solution Set

On a number line, draw an open circle at \(-3\) to indicate that \(-3\) is not included in the solution. Shade the region to the left of \(-3\), as this represents all values less than \(-3\).

Key Concepts

Inequality Solving

When faced with an inequality, the goal is to find all possible values of the variable that make the inequality true. In the exercise, we're given the inequality \( 7x + 3 < 9x - 3x \). The key steps are simplifying the inequality and isolating the variable.

First, simplify the expression. Combine like terms whenever possible. Here, the right side \( 9x - 3x \) simplifies to \( 6x \). Our inequality becomes \( 7x + 3 < 6x \).

Next, isolate the variable \( x \). Subtract \( 6x \) from both sides: \( 7x + 3 - 6x < 0 \). This simplifies to \( x + 3 < 0 \). Then, subtract 3 from both sides to solve for \( x \); giving us \( x < -3 \).

The result, \( x < -3 \), means that any number less than \(-3\) will satisfy the inequality.

Solution Set Notation

Writing an answer using solution set notation involves expressing exactly which values of the variable make the inequality true.

For the inequality \( x < -3 \), the solution set is precisely the collection of all numbers that are less than \(-3\). To express this in solution set notation, we use the format:

\[ \{ x \mid x < -3 \} \]

This reads as "the set of all \( x \) such that \( x \) is less than \(-3\)." Usage of solution set notation helps in displaying the results in a concise and mathematical manner, providing clarity on the set of numbers that solve the inequality.

Graphing Inequalities

Once you've determined the solution to an inequality, it is important to visually represent it through graphing. This process helps to quickly show which values satisfy the inequality.

For the inequality \( x < -3 \), begin by drawing a number line. Locate \(-3\) on the line and draw an open circle at this point. The open circle indicates that \(-3\) itself is not included in the solution set.

Next, shade the portion of the number line to the left of \(-3\). This shading represents all numbers that are less than \(-3\) and illustrates visually the values that make the inequality true.

Graphing inequalities not only aids in understanding the solution set but also provides a clear, visual context for analyzing solutions.