Problem 62 - Algebra 2

Question

Solve each inequality. (lessons \(1-5\) and \(1-6 )\) $$ |5-m| < 1 $$

Step-by-Step Solution

Answer

The solution is \(4 < m < 6\).

Step 1: Understanding Absolute Value Inequalities

The inequality given is \( |5-m| < 1 \). Absolute value inequalities of the form \(|x| < a\) are solved by creating a compound inequality: \[-a < x < a\] This states that the expression inside can range from \(-a\) to \(a\).

Step 2: Set Up the Compound Inequality

For the inequality \(|5-m| < 1\), we first rewrite it into a compound inequality based on the formula for absolute values: \[-1 < 5 - m < 1\].

Step 3: Solve the First Part of the Compound Inequality

The first part of the compound inequality is \(-1 < 5 - m\). To isolate \(m\), we subtract 5 from both sides: \[-1 - 5 < -m\] Simplifying, it becomes \[-6 < -m\]. Now, multiply by -1 and remember to reverse the inequality sign:\[m < 6\].

Step 4: Solve the Second Part of the Compound Inequality

The second part of the compound inequality is \(5 - m < 1\). Subtract 5 from both sides to isolate \(-m\): \[5 - 5 - m < 1 - 5\], which simplifies to \[-m < -4\]. Multiply by -1 and reverse the inequality sign: \[m > 4\].

Step 5: Combine the Inequalities

Now, combine the results from Step 3 and Step 4 to find the solution that satisfies both parts: \(4 < m < 6\). This is the range of values for \(m\) that satisfies the original inequality.

Key Concepts

Compound Inequalities

Compound inequalities are a fundamental concept in algebra where multiple inequalities are combined into a single statement. These inequalities typically involve two inequality signs and state that a variable lies between two values. For instance, when solving absolute value inequalities like \( |5-m| < 1 \), we turn it into a compound inequality. This results in two simultaneous inequalities, \[ -1 < 5 - m < 1 \], that solutions must satisfy both conditions.

To solve compound inequalities, you can break them into two separate inequalities:

• One for the lower bound
• Another for the upper bound

Both inequalities must be true for a solution to be valid. In our example, it means finding values of \( m \) that lie between these bounds.

Inequality Solving

Solving inequalities involves finding all possible values of the variable that make the inequality true. When dealing with absolute value inequalities, solving involves a few extra steps compared to basic linear inequalities.

Once we've established the compound inequalities, each part must be solved individually. For the inequality \(-1 < 5 - m\), we isolate \(m\) by performing arithmetic operations on both sides. This may include

• Adding or subtracting numbers
• Multiplying or dividing, remembering to reverse the inequality when multiplying or dividing by a negative number

For instance, reversing an inequality sign is crucial. It ensures the solution remains correct when dealing with negative coefficients or results.

Algebraic Manipulation

Algebraic manipulation is a set of skills used to rearrange equations or inequalities to isolate the variable and solve them effectively. It often involves operations such as addition, subtraction, multiplication, or division across both sides of the inequality.

In our example: \(|5-m| < 1\), we rewrote the inequality as \(-1 < 5 - m < 1\). To isolate \(m\), we changed this compound inequality into two simpler inequalities and used algebra to solve each.

• Subtracting values to shift terms across the inequality
• Multiplying or dividing by -1 when needed (and remembering to switch the inequality sign)

Through these manipulations, we determine \( 4 < m < 6 \) as the valid solution, demonstrating the value of clear and systematic algebraic work.

Range of Values

The range of values is the set of all numbers that satisfy an inequality. After solving a compound inequality, you are left with expressions that define these ranges.

In the example \(4 < m < 6\), the inequalities tell us that \( m \) must be greater than 4 but less than 6.
This interaction between two inequalities forms a clear range, often expressed with numbers or visualized using number lines. The understanding of what numbers satisfy the inequality is crucial in mathematics.

• It helps see the "big picture" – where solutions lie within certain constraints.
• Clarifies boundaries of variable values – ensuring solutions stay within designated limits.

Knowing the range of values is vital, especially when applying these solutions in real-world scenarios where precise boundaries matter.