Problem 58 - Elementary Algebra Within Reach

Question

In Exercises 55-58, solve and graph the inequality. $$ 3 x-7 \geq 2 x+9 $$

Step-by-Step Solution

Answer

The solution to the inequality \(3x - 7 \geq 2x + 9\) is \(x \geq 16\)

Step 1: Eliminate \(x\) on one side

To begin, let's get all terms with \(x\) on one side of the inequality and the numerical terms on the other. To do this, subtract \(2x\) from both sides. This results in \(3x - 2x \geq 2x + 9 - 2x\) which simplifies to \(x - 7 \geq 9\).

Step 2: Solve for \(x\)

Now, let's isolate \(x\) on one side of the inequality. To achieve that, add \(7\) to both sides. This results in \(x - 7 + 7 \geq 9 + 7\), which simplifies to \(x \geq 16\).

Step 3: Graph the solution

On the number line, a solid dot is used at number \(16\) and a line is drawn from \(16\) to the right with an arrow, indicating all numbers to the right of and including \(16\).

Step 4: Interpret the result

The solution \(x \geq 16\) means that all values of \(x\) that are equal to \(16\) or greater than \(16\) will satisfy the given inequality.

Key Concepts

Graphing Inequalities

When learning how to solve inequalities, a crucial concept to grasp is the process of graphing them. Graphing is a visual representation of all possible solutions to an inequality. For a simple linear inequality, such as the one in our exercise, \( x \geq 16 \), the graphing process begins with marking the critical value on the number line—in this case, the number 16.

To graph \( x \geq 16 \), first place a solid dot on 16, as the inequality includes the number itself (indicated by the 'equal to' component of the \geq symbol). The solid dot means that 16 is a part of the set of solutions. After placing the dot, draw a ray extending to the right, which shows that every number beyond 16 is also a solution. The use of an arrow indicates that the solutions continue indefinitely.

graphing inequalities involves understanding the symbols:

• \(>\) or \(<\) - a dashed line, indicating that the number at that point is not included in the solution set.
• \(\geq\) or \(\leq\) - a solid line, meaning the number at that point is included in the set of solutions.

In the context of our exercise, since the inequality symbol was \geq, we used a solid line to include 16 as it matched the inequality given.

Linear Inequalities

Linear inequalities, like the one presented in the exercise \(3x - 7 \geq 2x + 9\), are an extension of linear equations that one usually encounters in algebra. Instead of an equals sign, a linear inequality involves symbols that show the relationship between two expressions—specifically, whether one is larger, smaller or equal to the other.

The most common linear inequality symbols are:

• \(<\) - less than
• \(>\) - greater than
• \(\leq\) - less than or equal to
• \(\geq\) - greater than or equal to

To solve linear inequalities, we perform similar algebraic manipulations as with linear equations, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same nonzero number. However, it's pivotal to remember that if you multiply or divide both sides of an inequality by a negative number, the direction of the inequality must be reversed.

Once solved, a linear inequality like \( x \geq 16 \) tells us that x can take any value greater than or equal to 16. It's essential to write the final answer as \( x \geq 16 \) because it includes all numbers that satisfy the condition, not just a single solution.

Algebraic Manipulation

Algebraic manipulation is a cornerstone of solving various mathematical problems—particularly, solving inequalities. It involves the use of arithmetic operations and principles to rearrange and simplify equations and inequalities to find their solutions.

For example, in our exercise, the steps taken to isolate \(x\) and solve the inequality were as follows:

1. Subtract \(2x\) from both sides to eliminate \(x\) from one side of the inequality, resulting in \(x - 7 \geq 9\).
2. Add \(7\) to both sides to isolate \(x\), which gives us \(x \geq 16\).

During the course of algebraic manipulation, it's vital to maintain the balance of the inequality by performing the same operations on both sides. However, keep in mind that the rules of algebraic manipulation change slightly when dealing with inequalities compared to equations. Apart from the standard operations, flipping the inequality sign when multiplying or dividing by a negative number is a key rule to remember because it keeps the relational meaning of the inequality intact. Mastery of algebraic manipulation allows us to accurately solve and graph inequalities as seen in the exercise provided.