Problem 62 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ -x-3 \leq 4 $$

Step-by-Step Solution

Answer

The solution is \(x \geq -7\).

Step 1: Remove Constant Term

Add 3 to both sides of the inequality to remove the constant term from the left side: \[-x - 3 + 3 \leq 4 + 3\] which simplifies to \[-x \leq 7\].

Step 2: Solve for x

To isolate \(x\), divide both sides by -1. Remember, dividing by a negative number reverses the inequality sign:\[-x \leq 7\] becomes \[x \geq -7\].

Step 3: Verify the Solution

Check if the solution \(x \geq -7\) satisfies the original inequality. Choose a value, for instance \(x = -7\), and substitute it back:\[-(-7) - 3 \leq 4\]\[7 - 3 \leq 4\] \[4 \leq 4\]This is true, thus confirming the solution.

Key Concepts

Understanding Inequality Solutions

An inequality, like the one we have here, is similar to an equation but instead of using equality ("="), it expresses a relationship where one side may be smaller, larger, or equal under particular circumstances. The original inequality given is \[-x - 3 \leq 4\].

The "\(\leq\)" sign suggests that whatever solution we find must make the left-hand side less than or equal to the right-hand side. Solving inequalities involves similar steps to solving equations. The primary goal is to isolate the variable, usually represented by \(x\), on one side.

• The operations performed on inequalities mirror those in equations - we can add, subtract, multiply, or divide.
• However, one critical difference is: when we multiply or divide both sides by a negative number, the inequality sign must be flipped.

Understanding this fundamental concept allows us to approach and solve inequalities systematically.

Step-by-Step Solving Process

Breaking down the inequality into manageable steps simplifies the process:

Step 1: Remove Constant Term
We begin by eliminating the constant term from the side with the variable. In our example, we add 3 to both sides of the inequality:\[-x - 3 + 3 \leq 4 + 3\]. This results in \(-x \leq 7\).

Step 2: Solve for x
The next step is to isolate the variable, \(x\). Here, we need to divide by \(-1\) to solve for \(x\). Remember, dividing by a negative number reverses the inequality sign:\[-x \leq 7\] becomes \(x \geq -7\).

When you reverse the inequality sign, it reflects the change in relationship caused by multiplication or division through a negative number. This step-by-step solving approach ensures clarity and correctness in finding solutions to inequalities.

Verifying Solutions

Verification adds confidence to your solution by testing its validity. After obtaining \(x \geq -7\) as our solution, we should check if it is indeed correct. Here's how:

Pick a value that fits within the solution range. For instance, let's choose \(x = -7\), a boundary value.

• Substitute \(-7\) back into the original inequality \(-(-7) - 3 \leq 4\).
• Solve the simplified expression: \[7 - 3 \leq 4\].
• This statement is true because \(4 \leq 4\).

By verifying with at least one point from the solution set, we confirm the correctness of our solution \(x \geq -7\). Always verify to ensure that no step in the inequality solving process has been overlooked or misapplied.