Problem 61 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(15-(x+8) \geq 13\) (a) \(x=0\) (b) \(x=-6\) (c) \(x=2\) (d) \(x=-10\)

Step-by-Step Solution

Answer

The solutions of the inequality are \(x=-6\) and \(x=-10\).

Step 1: Understanding the inequality

Firstly, evaluate and simplify the given inequality. We have \(15-(x+8) \geq 13\), which simplifies to \(-x+7 \geq 13\). That further simplifies to \(x \leq -6\) after adding \(x\) and subtracting 13 from both sides.

Step 2: Setting each x value

Now, to find if each value of \(x\) is a solution of the inequality, substitute the values \(x=0\), \(x=-6\), \(x=2\), and \(x=-10\) into \(x \leq -6\).

Step 3: Checking \(x=0\)

If we substitute \(x=0\) into \(x \leq -6\), we obtain \(0 \leq -6\). This is not true, so \(x=0\) is not a solution of the inequality.

Step 4: Checking \(x=-6\)

If we substitute \(x=-6\) into \(x \leq -6\), we obtain \(-6 \leq -6\). This is true, so \(x=-6\) is a solution of the inequality.

Step 5: Checking \(x=2\)

If we substitute \(x=2\) into \(x \leq -6\), we obtain \(2 \leq -6\). This is not true, hence \(x=2\) is not a solution of the inequality.

Step 6: Checking \(x=-10\)

If we substitute \(x=-10\) into \(x \leq -6\), we obtain \(-10 \leq -6\). This is true, therefore \(x=-10\) is a solution of the inequality.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal to each other. They use symbols such as ">", "<", "≥", and "≤". In the exercise provided, our focus is on the inequality symbol "≥", which signifies "greater than or equal to."

When handling inequalities, it is crucial to maintain the direction of the inequality sign throughout operations. For example, if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This helps keep the relationships between values accurate.

So, ideas to remember include:

• Operate on both sides of an inequality similarly to equations, but be mindful of the inequality sign.
• Always flip the inequality sign when multiplying or dividing by a negative number.
• Check each step to ensure the inequality remains true with simplifications.

By understanding these basics, you can solve inequalities with confidence.

Algebraic Simplification

Algebra involves manipulating variables and constants to solve equations or inequalities. In our exercise, we start with the inequality expression: \[15 - (x + 8) \geq 13\].

When simplifying inequalities, the goal is to isolate the variable on one side.

Here are the steps performed in the simplification:

• Distribute any terms inside parentheses. For \(15 - (x + 8)\), expand it to \(15 - x - 8\).
• Simplify by combining like terms: the expression becomes \(-x + 7 \geq 13\).
• To isolate \(x\), first get rid of the constant term on the side with \(x\) by subtracting 7 from both sides, leading to \(-x \geq 6\).
• To solve for \(x\), multiply both sides by -1, which flips the inequality to \(x \leq -6\).

Algebraic manipulation like this is fundamental in rearranging expressions to make the inequality easier to interpret and solve.

Solution Verification

Solution verification in inequalities involves checking if given values satisfy the inequality condition. In the exercise, after simplification, we arrive at \(x \leq -6\). We need to verify this by substituting the given values of \(x\).

The process involves:

• Substituting each value into the simplified inequality.
• Evaluating if the statement holds true or not.

For example, by substituting \(x = 0\), we see \(0 \leq -6\) is false, thus \(x = 0\) is not a solution. Similarly, we verify other values:

• For \(x = -6\), the inequality becomes true, so it's a solution.
• For \(x = 2\), it results in false, not a solution.
• For \(x = -10\), it holds true, so it is a solution.

Solution verification is essential to ensure that our interpretations of the inequality are correct. It provides assurance that the values derived or given truly satisfy the conditions set forth by the inequality.

Step-by-Step Solutions

Step-by-step solutions are beneficial especially in complex problems like inequalities because they break down what can initially seem overwhelming into manageable actions. By following step-by-step methods, students can gain insights into the logical progression required to solve inequalities.

This approach involves:

• Identifying and understanding the problem or inequality presented.
• Step-by-step simplification by following algebraic rules.
• Verifying each proposed solution by checking validity within the inequality.

With each step concisely defined and executed, learners can approach problems methodically and reduce errors. Additionally, reviewing each step ensures comprehension and unveils the reasons behind each mathematical operation. In our example, the systematic approach helped verify which values satisfied the inequality, ensuring thorough understanding and clarity.