Problem 76 - Algebra 1

Question

Solve the inequality. Then sketch a graph of the solution on a number line. $$|3 x+7|-4>9 \quad $$

Step-by-Step Solution

Answer

The solution of the inequality is \( x > 2 \) or \( x < -\frac{20}{3} \).

Step 1: Isolate the absolute value expression

Add 4 to both sides of the inequality to isolate the absolute value term: |3x + 7| > 13.

Step 2: Rewrite as compound inequality

Now, rewrite the absolute value inequality as a compound inequality without absolute value symbols: either 3x + 7 > 13 or 3x + 7 < -13.

Step 3: Solve the compound inequality

For 3x + 7 > 13, subtract 7 from both sides and then divide by 3, we get: x > 2. For 3x + 7 < -13, subtract 7 from both sides and then divide by 3, we get: x < -20/3.

Step 4: Represent the solution on a number line

Draw a number line and mark points -20/3 and 2. The solution to the compound inequality is x > 2 or x < -20/3. This means solution includes all numbers greater than 2 and less than -20/3.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute value bars. The absolute value of a number is its distance from zero on a number line, regardless of direction. In this problem, the inequality \(|3x + 7| - 4 > 9\) requires solving for the condition where the magnitude is greater than a certain value.

To solve an absolute value inequality, it's pivotal to understand how to isolate the absolute value expression.

• Start by getting rid of any constants outside the absolute value, as shown by adding 4 to both sides of the inequality: \(|3x + 7| > 13\).
• Once isolated, you can express the statement as a set of two inequalities without the absolute value symbols: \(3x + 7 > 13\) or \(3x + 7 < -13\).

By translating the problem into these two parts, we create a path forward for solving the original inequality. Understanding this conversion is key for tackling any absolute value inequality effectively.

Compound Inequalities

Once you've rewritten an absolute value inequality as a compound inequality, you have two separate conditions to solve. A compound inequality contains two distinct inequalities joined by the words 'and' or 'or'. In this exercise, the rewritten form becomes two statements:

• \(3x + 7 > 13\)
• \(3x + 7 < -13\)

These types of compound inequalities rely on understanding when both parts need to be true ("and" scenarios) or just one needs to be true ("or" scenarios). In our case:

• "Or" indicates that if any one of the conditions is satisfied, the solution is valid.
• This translates to: \(x > 2\) or \(x < -\frac{20}{3}\).

By solving each inequality separately and considering the "or" logic, you're able to properly interpret the solution as a whole. Compound inequalities uniquely show where solutions can lie within two potential ranges.

Graphing on a Number Line

Once you've derived your solution, visualizing it on a number line solidifies your understanding. This involves marking specific key points and showing the regions that satisfy the inequality.Given \(x > 2\) or \(x < -\frac{20}{3}\), depict these on a number line:

• First, mark the points at \(x = 2\) and \(x = -\frac{20}{3}\) respectively.
• Shade or draw an arrow extending to the right from \(x = 2\) to indicate all numbers greater than 2.
• Likewise, shade or draw an arrow extending to the left from \(x = -\frac{20}{3}\) indicating all numbers less than \(-\frac{20}{3}\).

Using a number line offers a visual summary making it much easier to digest the range of possible solutions. Seeing each interval in perspective helps reinforce the understanding of how compound solution sets work.