Problem 45 - Prealgebra

Question

Solve each inequality. Check your solution. $$6 x<-27$$

Step-by-Step Solution

Answer

The solution is \(x < -4.5\).

Step 1: Isolate the variable

To solve the inequality \(6x < -27\), we first need to isolate \(x\). To do this, divide both sides of the inequality by 6. This results in \(x < \frac{-27}{6}\).

Step 2: Simplify the inequality

Simplify the fraction \(-\frac{27}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This results in \(x < -\frac{9}{2}\) or \(x < -4.5\).

Step 3: Check the solution

To verify the solution, choose a number less than \(-4.5\), such as \(-5\), and substitute it back into the original inequality. Check if the inequality holds: \(6(-5) = -30 < -27\), which is true. This confirms that the solution is correct.

Key Concepts

Isolating the Variable

When solving inequalities, one of the essential first steps is to isolate the variable. In our exercise, we are dealing with the inequality \(6x < -27\). Our goal is to get \(x\) by itself on one side of the inequality.
To achieve this, we'll perform the same operation on both sides of the inequality, maintaining its balance. Since \(x\) is being multiplied by 6, the way to isolate \(x\) involves dividing both sides by 6. This operation results in \(x < \frac{-27}{6}\). By isolating the variable in this manner, we've taken a crucial step towards solving the inequality.
Remember, dividing or multiplying both sides of an inequality by a positive number does not change the inequality sign. However, if you divide or multiply by a negative number, be sure to flip the inequality sign.

Simplifying Fractions

Simplifying fractions is an important skill when solving equations and inequalities, as it makes numbers easier to understand and compare. In our case, after dividing both sides of the inequality by 6, we've arrived at \(x < \frac{-27}{6}\).
Now, let's simplify \(-\frac{27}{6}\). Start by finding the greatest common divisor (GCD) of 27 and 6. The GCD is 3, since 3 is the largest number that divides both numbers without leaving a remainder.

• Divide the numerator and the denominator by the GCD:
• \(\frac{-27}{3} = -9\)
• \(\frac{6}{3} = 2\)

After performing the simplification, we get \(x < -\frac{9}{2}\), which can also be expressed as \(x < -4.5\). This makes our final inequality clearer and easier to interpret.

Checking Solutions

After simplifying your inequality, it's vital to check your solution to ensure its accuracy. This step ensures that no calculation mistakes were made. To verify our solution, \(x < -4.5\), choose a value that satisfies this condition. For example, let's use \(-5\), which is less than \(-4.5\).Plug this value back into the original inequality, which is \(6x < -27\). Substitute \(-5\) into the place of \(x\):

• \(6 \times (-5) = -30\)
• Next, check if \(-30 < -27\) is true.

Since \(-30 < -27\) is indeed true, this confirms that \(x = -5\) is a valid solution. Hence, our solution \(x < -4.5\) is correct. Checking solutions helps reinforce your understanding and ensures the reliability of your answer.