Problem 1 - Prealgebra

Question

Solve each inequality. Check your solution. $$2 x<8$$

Step-by-Step Solution

Answer

The solution is \(x < 4\).

Step 1: Understand the inequality

The inequality given is \(2x < 8\). The goal is to find the values of \(x\) for which this inequality holds true.

Step 2: Isolate the variable

To solve \(2x < 8\), divide both sides of the inequality by 2 to isolate \(x\). This gives \(x < \frac{8}{2}\).

Step 3: Simplify the right side

Simplify \(\frac{8}{2}\) to obtain \(x < 4\).

Step 4: Interpret the solution

The solution \(x < 4\) means that any value of \(x\) that is less than 4 satisfies the inequality. This implies that the solution is the interval \((-\infty, 4)\).

Step 5: Check the solution

Choose a number less than 4, for instance, \(x = 3\). Substitute it back into the original inequality: \(2(3) < 8\). This simplifies to \(6 < 8\), which is true, confirming our solution. Similarly, if you choose a number equal to or greater than 4, the inequality will not hold.

Key Concepts

Inequality Isolation Steps

To solve an inequality such as \(2x < 8\), one critical approach is the isolation of the variable. The aim is to have \(x\) by itself on one side of the inequality sign. Here's a step-by-step guide on how to isolate \(x\):

• Start with the inequality: \(2x < 8\).
• Perform operations that simplify the equation. In this case, divide both sides by 2 to cancel out the coefficient of \(x\).
• After dividing, you’ll get \(x < \frac{8}{2}\).
• Simplify further to reveal \(x < 4\). This means all values of \(x\) that are less than 4 satisfy the inequality.

Through these steps, isolating the variable gives us a clear and succinct answer.

Checking Inequality Solutions

Once you've isolated the variable and found a solution, it's important to check whether the solutions indeed satisfy the original inequality. This verification step is simple yet crucial:

• Choose a value from your solution set. If \(x < 4\), select any number less than 4, say \(x = 3\).
• Substitute this number back into the original inequality: \(2(3) < 8\).
• Calculate the left side to check if the inequality holds: \(6 < 8\), which is true.
• Optionally, try a number not in the solution set, like \(x = 4\). Here, \(2(4) = 8\), which does not satisfy the inequality \(2x < 8\).

Confirming the solution ensures that the steps taken are correct and the solution is valid.

Interpreting Inequality Solutions

Interpreting the solution of an inequality involves understanding what the solution set represents. Once you obtain \(x < 4\), here's how to interpret it effectively:

• View the inequality as an interval: \(( -\infty, 4)\) represents all real numbers less than 4.
• Graphically, you can illustrate this on a number line, shading to the left of 4 and using an open circle at 4 to show it’s not included.
• In word problems, relate it to the context given. For example, if \(x\) represents the maximum number of items you can buy, it would mean you can buy fewer than 4 items.

Understanding the solution set helps in applying inequalities to real-world scenarios and ensures comprehensive problem-solving.