Problem 58 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 3-x \geq 4 $$

Step-by-Step Solution

Answer

to \(-1\).

Step 1: Understand the inequality

In this step, we will understand the given inequality: $$ 3 - x \geq 4 $$ The objective is to find the values of \(x\) for which this inequality holds true.

Step 2: Isolate the variable (x)

We want to isolate \(x\) on one side of the inequality. We'll start by subtracting \(3\) from both sides: $$ - x \geq 1 $$

Step 3: Negate both sides of the inequality

We want to get rid of the negative sign in front of \(x\). We can do this by multiplying or dividing both sides of the inequality by \(-1\). Remember that if you multiply or divide by a negative number, you must flip the inequality sign: $$ x \leq -1 $$

Step 4: Write the solution

The inequality is now simplified with \(x\) isolated: $$ x \leq -1 $$ This means that the solution to the inequality is all values of \(x\) less than or equal%2

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions involving variables where the two sides are not necessarily equal, which is indicated by inequality symbols such as greater than (>) or less than (<). Unlike equations, where both sides must equal, inequalities show a relationship where one side is greater or less than the other. This concept extends to 'greater than or equal to' (\textgreater=) and 'less than or equal to' (\textless=).
In the context of the provided exercise \(3 - x \textgreater= 4\), the goal is to find all the values of \(x\) that make the inequality true. Understanding how to manipulate these inequalities while maintaining the relationships between both sides is a fundamental algebra skill. This skill provides insight into the range of possible solutions rather than a single numerical answer, making it a crucial tool in many real-world applications such as comparing statistics, budgeting, and scientific measurements.

Inequality Notation

Inequalities are expressed using specific notations which indicate the nature of the relationship between the values or expressions involved. The key notations include:

• Less than (\textless): Indicates that the value on the left side is smaller than the value on the right side.
• Greater than (\textgreater): The left side value is larger than the right side value.
• Less than or equal to (\textless=): The left side value is either less than or exactly equal to the right side value.
• Greater than or equal to (\textgreater=): The left-side value is greater than or precisely equal to the right side.

When these notations are used, it's important to know that the inequality's direction can change under certain operations, such as when multiplying or dividing by a negative number. This principle is crucial to solve inequalities correctly, as seen in step 3 of the exercise where multiplying by \( -1 \) flipped the inequality from \( -x \textgreater= 1 \) to \( x \textless= -1 \).

Isolating Variables

Isolating the variable is a core process in solving both equations and inequalities. It involves manipulating the algebraic expression so that the variable of interest is on one side of the equation or inequality, and everything else is on the other side. The primary goal is to express the variable in terms of known quantities or other variables.
In doing so, one typically performs operations such as adding, subtracting, multiplying, or dividing both sides by the same number, except in the case of inequalities involving multiplication or division by a negative number, this leads to a switch in the inequality direction. For the exercise \(3 - x \textgreater= 4\), subtracting 3 from both sides gives \( -x \textgreater= 1\) and then dividing by \( -1 \) and reversing the inequality gives the solution \( x \textless= -1\). These steps highlight the importance of understanding how to isolate variables to effectively solve algebraic inequalities.