Problem 86 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ 7 \geq|15 x-45|+7 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\), represented as the interval \([3, 3]\).

Step 1: Isolate the Absolute Value

Begin by isolating the absolute value expression on one side of the inequality. We will subtract 7 from both sides of the inequality \(7 \geq |15x - 45| + 7\): \[ 7 - 7 \geq |15x - 45| \]\[ 0 \geq |15x - 45| \]

Step 2: Solve the Absolute Value Inequality

Since the absolute value \(|15x - 45|\) is always non-negative, it can only be less than or equal to zero when it is exactly zero. Hence, set the expression inside the absolute value equal to zero:\[ 15x - 45 = 0 \]

Step 3: Solve for x

Solve the equation from Step 2 for \(x\). Add 45 to both sides:\[ 15x = 45 \]Now, divide both sides by 15:\[ x = 3 \]

Step 4: Represent the Solution

The solution to our inequality is \(x = 3\) because that's the only point where the expression inside the absolute value equals zero. The graph is just the single point \(x = 3\). In interval notation, we represent this as:\[ [3, 3] \]

Key Concepts

Isolating the Absolute Value

When tackling equations or inequalities involving absolute values, a key step is to isolate the absolute value expression. This means you need to get the absolute value term by itself on one side of the equation or inequality. Consider the original inequality:

\[ 7 \geq |15x - 45| + 7 \]
To isolate the absolute value, subtract 7 from both sides. This cancels out the +7 added to the absolute value on the right side.

This subtraction simplifies to:
\[ 0 \geq |15x - 45| \]
Now, the absolute value is isolated. At this point, it’s crucial to understand that the expression inside the absolute value can only be zero or positive.

Graphing Solutions

Once the absolute value has been isolated, the next step is to understand how to find and graph the solution. The inequality \(0 \geq |15x - 45|\) implies that the absolute value expression equals zero because absolute values can't be negative.

So, set the expression inside the absolute value equal to zero:
\[15x - 45 = 0\]
Solve for \(x\) to find the critical point that will be graphed. Solving gives \(x = 3\).

Geometrically, this solution is represented graphically just by the single point \(x = 3\) on a number line. Since we're dealing with an equation where the absolute value expression equals zero, our graph is a dot at this precise point, as there is only one solution.

Interval Notation

Interval notation is a shorthand way of writing sets of numbers, commonly used to describe solutions to inequalities. It makes it easy to see which numbers are included in the solution set.

For this specific problem, where we find \(x = 3\) as the only solution, interval notation allows us to compactly write the solution as \([3, 3]\).

• The square brackets, \([\ ]\), indicate that the endpoint is included in the set, meaning \(x = 3\) is a exact point in the solution set.
• Here, both the lower and upper endpoints of the interval are the same, denoting that the solution is just a single point.

Using interval notation in this way provides a clear and concise way to express solutions involving absolute values, especially when they have endpoints or are specific points on a graph.