Problem 40 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ 3-|2 x+4| \leq 1 $$

Step-by-Step Solution

Answer

The solution is \(( -\infty, -3 ] \cup [ -1, \infty )\).

Step 1: Isolate Absolute Value

The inequality is \(3 - |2x + 4| \leq 1\). Start by isolating the absolute value expression: subtract 3 from both sides to get \(-|2x + 4| \leq -2\). Multiply every term by -1 to obtain \(|2x + 4| \geq 2\), remembering to reverse the inequality sign when multiplying by a negative number.

Step 2: Remove Absolute Value

The inequality \(|2x + 4| \geq 2\) means that the expression inside the absolute value could be either greater than or equal to 2 or less than or equal to -2. So, split into two inequalities: \(2x + 4 \geq 2\) and \(2x + 4 \leq -2\).

Step 3: Solve First Inequality

Solve \(2x + 4 \geq 2\) by subtracting 4 from both sides: \(2x \geq -2\). Then, divide both sides by 2 to get \(x \geq -1\).

Step 4: Solve Second Inequality

Solve \(2x + 4 \leq -2\) by subtracting 4 from both sides: \(2x \leq -6\). Then, divide both sides by 2 to get \(x \leq -3\).

Step 5: Combine Solutions

The solutions \(x \geq -1\) and \(x \leq -3\) are combined because \(|2x + 4| \geq 2\) indicates the values of \(x\) fall outside the range between these numbers. Therefore, the interval notation solution is \(( -\infty, -3 ] \cup [ -1, \infty )\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities can seem challenging at first, but with a clear understanding of the concept, they become much easier to handle. Absolute value refers to the distance of a number from zero on the number line, regardless of direction. Therefore, absolute value inequalities involve expressions within absolute value bars that need to satisfy a certain condition.

When you face an absolute value inequality like \(|a| \geq b\), it implies the expression inside the absolute value can be either greater than or equal to \(b\) or less than or equal to \(-b\). This is due to the nature of absolute value capturing two possible scenarios on the number line:

• One where the expression is a positive distance (\(a \geq b\)).
• Another where it is a negative distance towards zero (\(a \leq -b\)).

By understanding this, you can break down the inequality into two simple linear inequalities and solve them independently.

Interval Notation

Interval notation is a helpful way to express the range of solutions for inequalities in a concise form. It provides a clear visual of the solution set that can be easily understood. This notation uses brackets and parentheses:

• Square brackets \([\, ]\) indicate that an endpoint is included in the solution (\(\leq\) or \(\geq\)).
• Parentheses \((\, )\) denote that the endpoint is not included (\(<\) or \(>\)).

Using interval notation, solutions are written like intervals on a number line. For example, the solution to an inequality \(x \geq -1\) would be written in interval notation as \([-1, \infty)\).

In the provided exercise, the interval notation \((-\infty, -3] \cup [-1, \infty)\) expresses two separate ranges - the first interval representing all values from \(-\infty\) to \(-3\) (including \(-3\)), and the second covering values from \(-1\) to \(\infty\), starting at \(-1\). The union sign \(\cup\) is used to combine these intervals, showing that either range is part of the solution.

Inequality Solution Steps

Solving inequalities is a systematic endeavor. Let's break down the steps used in our example to solve an absolute value inequality. The process can generally be structured as follows:

• **Isolate the Absolute Value**: Begin by isolating the absolute value on one side of the inequality. This involves rearranging the inequality if necessary, such as adding or subtracting terms from both sides.
• **Flip and Split**: Once isolated, interpret the absolute value inequality by reversing and creating two separate inequalities. Remember, multiplying through by a negative number reverses the inequality sign.
• **Solve Each Inequality**: Work through each of the simpler inequalities independently. Use basic algebraic manipulation like adding, subtracting, multiplying, or dividing to find the solution sets for each.
• **Combine Solutions**: Finally, evaluate whether to union the solutions, which means joining different solution sets using the union symbol \(\cup\), particularly when dealing with absolute value inequalities that create non-overlapping solution regions.

By adhering to these steps, you can navigate the complexity of inequalities confidently and accurately. Always ensure to check your final solution against both inequalities to confirm correctness.