Problem 21 - Algebra 1: Concepts and Skills

Question

Solve the inequality. $$ -6+5 x<19 $$

Step-by-Step Solution

Answer

Under the given conditions, x is less than 5.

Step 1: Re-arrange the inequality

Add 6 to each side of the inequality to isolate the term with x on one side. This results in \(5x < 25\).

Step 2: Isolate the variable

Divide each side of the inequality by 5 to solve for x. This gives \(x < 5\).

Key Concepts

Inequality Manipulation

When faced with an inequality, such as \( -6+5x<19 \), the goal is to isolate the variable and solve for it, much like you would with an equation. However, with inequalities, the direction of the inequality sign can change based on the operations you perform. In this example, the first step is to simplify and re-arrange the inequality to move closer to isolating the variable \(x\).

You'd begin by performing the inverse operation to the constant term on the side with the variable. In the exercise, to eliminate the \( -6 \), we add 6 to both sides of the inequality: \( -6 + 5x + 6 < 19 + 6 \). Simplifying this gives us \( 5x < 25 \), where the variable is closer to being isolated. It's crucial to perform the same operation on both sides to maintain the balance of the inequality. Always remember, if you multiply or divide by a negative number, the inequality sign flips. It's like a caution sign on this math road trip - watch out for it!

Isolating Variables

Isolating the variable in an inequality is akin to finding the key to unlock a door. The variable represents the unknown, and the inequality defines the range in which this unknown lies. To find the value of \(x\) in our inequality \(5x < 25\), we need to perform an operation that will 'free' \(x\) from any coefficients or constants attached to it.

In this case, since \(x\) is being multiplied by 5, the inverse operation would be to divide by 5. Applying this to both sides of the inequality, we get \(\frac{5x}{5} < \frac{25}{5}\). Simplifying both sides leads to \(x < 5\). This final expression is our key; it tells us that \(x\) can be any real number less than 5. This step is critical and allows us to understand clearly and concisely the range of solutions that satisfy the original problem.

Linear Inequalities

Linear inequalities are similar to linear equations but, instead of an equal sign, they have an inequality sign such as <, \(>\), \(\leq\), or \(\geq\). They're part of a basic toolkit in algebra that paints a broad picture of possible solutions, rather than pinpointing a single answer. The inequality we've been working with, \(x < 5\), is linear because it can be graphed as a straight line when using a different equation (\(y\)) to represent all possible values under 5.

To understand solutions to linear inequalities, it's also helpful to think about number lines. For \(x < 5\), imagine a number line with every point to the left of 5 shaded, extending indefinitely. Those points represent all the possible values for \(x\) that satisfy the inequality. Unlike equations, where there may be just one solution, inequalities describe a range of solutions. This understanding is fundamental when moving onto more complex tasks involving inequalities such as systems or those with absolute values.