Problem 100 - Algebra and Trigonometry

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $$ y=|2 x-5|+1 \text { and } y>9 $$

Step-by-Step Solution

Answer

Therefore, the values of \(x\) that solve the inequality are all \(x\) that belong to the interval \((-∞, -1.5) \cup (6.5, ∞)\).

Step 1: Work with the inequality

We're given that \(y = |2x - 5| + 1\) and \(y > 9\). Let's substitute the value of \(y\) in the inequality from the equation. So we get \(|2x - 5| + 1 > 9\). Now subtract 1 from both sides. We get \(|2x - 5| > 8\).

Step 2: Break down the absolute value

The absolute value \( |a| \) has two possible solutions: \( a \) and \( -a \). So we can rewrite our inequality as two separate inequalities \( 2x - 5 > 8 \) and \( 2x - 5 < -8 \).

Step 3: Solve the inequalities

For each inequality, we will isolate \(x\), which will give us the range of values it can take. We solve the first inequality and arrive at \(x > 6.5\), and from the second, we get \(x < -1.5\).

Step 4: Combine the ranges

Given how the ranges for \(x\) are obtained, there's no value \(x\) that simultaneously obeys \(x > 6.5\) and \(x < -1.5\), hence there's no intersection. Thus, our answer will contain two separate ranges.

Key Concepts

Absolute Value Inequality

Understanding absolute value inequalities can greatly simplify the process of finding solutions. Absolute value refers to the distance a number is from zero on the number line, regardless of direction.
Thus, the absolute value of any real number is always non-negative.When solving an absolute value inequality like \( |2x - 5| > 8 \), it's important to remember that it involves considering two scenarios:

• \( 2x - 5 > 8 \): Here, the expression inside the absolute value is greater than 8.
• \( 2x - 5 < -8 \): Here, the expression is less than -8.

These scenarios arise because an absolute value represents two possible situations depending on the direction from zero. By splitting the inequality into these two parts, you resolve the absolute value, making the problem easier to understand and solve.

Compound Inequalities

Compound inequalities involve two separate inequalities joined by the words 'and' or 'or'.
These words determine how the intervals are expressed in notation.In the case of the problem statement with \( |2x - 5| > 8 \), you effectively have two separate inequalities to solve:

• \( 2x - 5 > 8 \) which simplifies to \( x > 6.5 \)
• \( 2x - 5 < -8 \) which simplifies to \( x < -1.5 \)

These results form a compound inequality using 'or', meaning that the solution does not have any intersection. Each inequality stands alone, providing its own valid set of values for \( x \).
Understanding this concept is crucial as it dictates when a solution set will occupy a continuous range, or when it will segregate into two non-overlapping ranges.

Solving Inequalities

Solving inequalities follows a path similar to solving equations, with additional considerations.
An inequality shows a relationship where expressions are not necessarily equal, often using symbols like >, <, ≥, or ≤. When solving inequalities such as \( 2x - 5 > 8 \) or \( 2x - 5 < -8 \), the key is to isolate \( x \).

• Add or subtract terms on both sides to eliminate constant terms.
• Divide or multiply to remove coefficients of \( x \).

For example, by adding 5 to both sides of the first inequality and then dividing by 2, we find that \( x > 6.5 \).
The same operations for the second inequality yield \( x < -1.5 \).Finally, always be aware that multiplying or dividing both sides of an inequality by a negative number will flip the inequality sign. Always check your work to ensure the inequality relationship holds true throughout the process.