Problem 21 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 3 x-7 \leq 8 $$

Step-by-Step Solution

Answer

Answer: The solution for the inequality is \(x \leq 5\).

Step 1: Isolate the term with the variable x

Add 7 to both sides of the inequality to isolate the term with the x: $$ 3x - 7 + 7 \leq 8 + 7 $$ This simplifies to: $$ 3x \leq 15 $$

Step 2: Solve for x

Now, we will solve for x by dividing both sides of the inequality by 3: $$ \frac{3x}{3} \leq \frac{15}{3} $$ And this gives us the solution for x: $$ x \leq 5 $$ The solution to the inequality \(3x-7 \leq 8\) is \(x \leq 5\).

Key Concepts

Isolation of Variable

When solving inequalities like \(3x - 7 \leq 8\), it's crucial to start by isolating the variable, which in this case is \(x\). This process means getting \(x\) by itself on one side of the inequality sign.
To continue, you'll want to focus on the term containing \(x\)—in this inequality, it's \(3x\).
By adding \(7\) to both sides of the inequality, you remove the constant \(-7\) from the left side and balance the inequality:

• Original inequality: \(3x - 7 \leq 8\)
• Add \(7\): \(3x - 7 + 7 \leq 8 + 7\)
• This simplifies to: \(3x \leq 15\)

The importance of isolation is that it sets the stage for easier manipulation of equations and clear solutions.

Algebraic Manipulation

Algebraic manipulation is the technique of rearranging and simplifying expressions. This process is critical when you attempt to solve inequalities.
After isolating the variable term, as we did with \(3x \leq 15\), algebraic manipulation allows you to solve for \(x\) by performing arithmetic operations like addition, subtraction, multiplication, and division on both sides of the inequality.
In our example, dividing both sides of the inequality by \(3\) simplifies the expression:

• Divide by \(3\): \(\frac{3x}{3} \leq \frac{15}{3}\)
• This results in: \(x \leq 5\)

This step involves key skills of balancing both sides and ensuring equality remains true. It's all about precise calculation and maintaining the integrity of the inequality.

Inequality Solution

Solving an inequality is about finding all possible values of a variable that make the inequality true.
In our example, this involves the steps of isolation and algebraic manipulation. Once \(x\) is isolated, you can determine the acceptable range.
For \(3x - 7 \leq 8\), after simplifying, we found \(x \leq 5\). This tells us that any number \(x\) equal to or less than \(5\) satisfies the inequality.
The solution is denoted as \(x \leq 5\), meaning:

• \(x\) can be any value from negative infinity up to \(5\)
• Graphically, this is shown as a line with a closed circle at \(5\) and shaded to the left

Understanding inequality solutions helps in interpreting data ranges and possible scenarios in real-world problems.