Problem 14 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ -3 a+1<-5 $$

Step-by-Step Solution

Answer

Question: Solve the inequality -3a + 1 < -5 and determine the solution. Answer: a > 2

Step 1: Subtract 1 from both sides of the inequality

Firstly, subtract 1 from both sides of the inequality to eliminate the constant term on the left side. So, the inequality will become: $$-3a + 1 - 1 < -5 -1$$ $$-3a < -6$$

Step 2: Divide both sides of the inequality by -3

Now, divide both sides of the inequality by -3 to isolate the variable \(a\). Remember that when you multiply or divide both sides of an inequality by a negative number, the inequality sign must be flipped. Therefore, the inequality becomes: $$\frac{-3a}{-3} > \frac{-6}{-3}$$ $$a > 2$$ The solution to the inequality, \(-3a + 1 < -5\), is \(a > 2\).

Key Concepts

Understanding Algebra in Inequalities

Algebra is essential for solving inequalities like the given problem, \(-3a + 1 < -5\). It's all about manipulating expressions to find a value or set of values that satisfy a condition.
In this context, algebra helps rearrange the inequality to isolate the unknown variable, \(a\).

Here's how algebra is applied to this problem:

• First, identify the terms that need to be simplified or moved to solve for the variable \(a\).
• Next, perform operations on both sides of the inequality to maintain balance.

In step one, we subtracted 1 from both sides.This action utilizes algebraic properties to simplify one side of the inequality.
By doing so, we make it easier to later find what \(a\) must be greater than.
With equations and inequalities alike, algebra is the toolbox for rearranging terms and solving for variables.

Inequality Reversal and its Importance

When dealing with inequalities, there's a critical rule to remember: reversing the inequality sign when multiplying or dividing by a negative number. This rule was crucial in step two of the solution when dividing by \(-3\).

Here's why it's important:

• In step two, the inequality \(-3a < -6\) was divided by \(-3\) to isolate \(a\).
• The division of a negative switches the inequality from \(<\) to \(>\). If this step is missed, it leads to incorrect results.

An easy way to remember this is that negative operations "flip" the inequality. Imagine flipping a sign to keep the inequality true.

Understanding this principle means you won't make a common error and ensures accurate solutions every time!

Solving with Linear Equations

Linear equations form the backbone of many problems in algebra, including inequalities. Solving inequalities often involves manipulating expressions similar to a standard linear equation.
Here’s how linear equations align with solving inequalities like \(-3a + 1 < -5\):

• Linear equations are solved by isolating the variable on one side, often using algebraic operations such as subtraction or division.
• For instance, in our example, the subtraction of 1 is akin to simplifying terms in a linear equation \( ax + b = c \).
• After isolating the variable, dividing by a coefficient (the number multiplying the variable) is typically the last step.

While linear equations and inequalities share techniques, inequalities require careful attention to detail, especially with inequality reversal.
Mastering these steps paves the way for tackling more complex algebraic challenges.