Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. $$2 x+9 \leq x+2$$

Step-by-Step Solution

Answer

\(x \leq -7\). The graph would be a filled dot on -7 with a line extending to the left.

Step 1: Simplify the inequality

Start by subtracting \(x\) from both sides of the inequality to isolate \(x\) terms on one side: \(2x - x + 9 \leq x - x + 2\), which simplifies to \(x + 9 \leq 2\)

Step 2: Solve for x

Next, subtract 9 from both sides of the inequality to isolate \(x\): \(x + 9 - 9 \leq 2 - 9\), which simplifies to \(x \leq -7\)

Step 3: Graphing the solution on a number line

Plot a number line and mark -7 on it. Because the inequality sign \(\leq\) includes the possibility of \(x\) being equal to -7, fill in the dot on -7. Then, draw a line extending to the left from -7 to indicate that all numbers less than -7 are part of the solution set

Key Concepts

Addition Property of Inequality

The addition property of inequality states that you can add the same number to both sides of an inequality without changing the inequality's direction. For example, if you have an inequality like \( a < b \), adding \( c \) to both sides retains the truth of the inequality, so \( a + c < b + c \). This property is essential when solving inequalities because it allows us to isolate the variable we're solving for by eliminating constants or other variables on one side.

In the case of our exercise, we initially have \( 2x + 9 \leq x + 2 \). To isolate the \( x \) terms on one side using the addition property, we subtract \( x \) from both sides, which gives us \( x + 9 \leq 2 \). We continue using the addition property by subtracting 9 from both sides to completely isolate \( x \), leading to \( x \leq -7 \). These steps move us closer to understanding which values of \( x \) will satisfy the initial inequality.

Graphing on a Number Line

After algebraically manipulating an inequality to find the solution for the variable, the next step is to represent this solution visually using a number line. Graphing on a number line gives an intuitive understanding of all the possible values that satisfy the inequality.

For the inequality \( x \leq -7 \), we place a filled dot on -7 to show that -7 is included in the solution set due to the '\(\leq\)' sign, indicating 'less than or equal to.' Next, we draw a line or arrow pointing to the left starting from the value -7 to represent all numbers less than -7. This tells us that every number on the number line that is to the left of -7 is also a solution of the inequality. Graphing makes inequalities tangible, showing the range of values on the familiar context of the number line.

Algebraic Manipulation

Algebraic manipulation involves reorganizing or simplifying algebraic expressions or inequalities to make them easier to solve. This often involves using a series of algebraic properties, such as the addition property of inequality, the distributive property, and combining like terms.

In our exercise, the first step was to subtract \( x \) from both sides of the inequality in order to get the variable terms on one side. This is an example of algebraic manipulation that simplifies the inequality to \( x + 9 \leq 2 \). We performed another manipulation by then subtracting 9 from both sides, which isolated the variable and gave us the answer, \( x \leq -7 \). Algebraic manipulation is a cornerstone of solving equations and inequalities alike, providing the means to resume from complex relationships to simple, solvable conditions.