Problem 4 - Algebra Form and Function

Question

Are there values of \(x\) which satisfy the statements? Explain how you can tell without finding, or attempting to find, the values. $$ |2 x+5|=|-7-10| $$

Step-by-Step Solution

Answer

Yes, there are values of x that satisfy the given equation. We can determine this by recognizing the properties of absolute values and considering two cases: one for the positive value and one for the negative value inside the absolute value.

Step 1: Recognize the properties of absolute values

Absolute value is the distance of a number from 0, and it is never negative. It can be written as: $$ |a| = \begin{cases} a, \text{ if } a \ge 0 \\ -a, \text{ if } a < 0 \end{cases} $$ From this definition, it is clear that \(|a|=|-a|\) because the distance from 0 is the same for \(a\) and \(-a\).

Step 2: Consider the given absolute values

We are given the equation: $$ |2 x+5|=|-7-10| $$ First, let's simplify the equation. Since \(|-7-10|=|-17|=17\), the equation becomes: $$ |2 x+5|=17 $$

Step 3: Determine if there is a solution for x

Now imagine two cases, one for positive value inside the absolute value and one for negative: $$ 2x + 5 = 17 \\ 2x + 5 = -17 $$ In both cases, a real value of x exists that will satisfy the equation. For the first case, \(x=6\), while for the second case, \(x=-11\). So, there are values of \(x\) that satisfy the given equation.

Step 4: Conclusion

Yes, there are values of \(x\) which satisfy the given equation. Without actually finding the values of \(x\), we could determine that there are solutions by recognizing the properties of absolute values and showing two cases, one for positive value and one for negative value.

Key Concepts

Properties of Absolute Value

Absolute value can seem a bit tricky since it involves how far a number is from zero, regardless of direction. Simply put, the absolute value of a number is always non-negative. This is because it represents a distance, which can't be negative.

When you see an expression like \(|a|\), think of it as asking "how far is \(a\) from zero?" Consider the crucial properties of absolute values:

• If \(a\) is positive or zero, then \(|a| = a\). If \(a\) is negative, then \(|a| = -a\).
• Absolute value ensures that \(|a| = |-a|\) because they are both the same distance away from zero.

In our exercise, these properties allow us to equate two different expressions by their absolute values and simplify without worrying about signs. This simplification process is crucial when solving equations that involve absolute values.

Solving Equations

When solving equations, especially those with absolute values, keep in mind that these involve considering different scenarios. With an expression like \(|2x + 5| = 17\), you must consider two possible equations because you can have two different cases.

First, consider when what's inside the absolute value is positive: \(2x + 5 = 17\). Solve it as you would a linear equation, finding \(x = 6\).

• Subtract 5 from both sides: \(2x = 12\).
• Divide by 2: \(x = 6\).

Next, consider when the expression inside the absolute value is negative: \(2x + 5 = -17\). Again, solve as a regular equation, leading to \(x = -11\).

• Subtract 5 from both sides: \(2x = -22\).
• Divide by 2: \(x = -11\).

By examining both cases, we find possible solutions that satisfy the original equation with an absolute value.

Real Values of x

When you solve equations involving absolute values, you find the real values of \(x\) that make the equation true. These values make the mathematical statement hold in the "real world" of numbers.

In the equation \(|2x + 5| = 17\), the solutions are real numbers where the expression inside the absolute value equals either \(17\) or \(-17\). These solutions, \(x = 6\) and \(x = -11\), are both real numbers, meaning they exist on the number line without any imaginary component.

This exercise demonstrates that absolute value equations often have two solutions because of their nature of measuring distance. Recognizing this pattern helps you confidently approach similar problems and determine whether certain equations have real solutions.