Problem 13 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ -11 < -4 x+5 < 13 $$

Step-by-Step Solution

Answer

The solution is \(-2 < x < 4\).

Step 1: Isolate the Middle Part of the Inequality

Start by subtracting 5 from all parts of the inequality to simplify it. This step reduces the inequality from \(-11 < -4x+5 < 13\) to \(-16 < -4x < 8\).

Step 2: Solve for x

Divide all parts of the inequality by -4 to solve for \(x\), remembering to reverse the inequality signs because we are dividing by a negative number. This gives us \(4 > x > -2\) or equivalently \(-2 < x < 4\).

Step 3: Graph the Solution Set

Draw a number line and mark open circles at \(-2\) and \(4\) to indicate that these endpoints are not included in the solution. Shade the region between \(-2\) and \(4\) to represent all the values \(x\) can take.

Key Concepts

Graphing on a Number Line

Graphing solutions on a number line helps us visually understand where our values lie. In this exercise, once we have found the solution to the inequality, \(-2 < x < 4\), the next step is to represent this solution on a number line. This makes it easy to see all the possible values that \(x\) can take.

Here’s how to graph this on a number line:

• Draw a horizontal line and mark it with numbers, ensuring to include the critical points \(-2\) and \(4\).
• Place open circles at \(-2\) and \(4\) since these numbers are not part of the solution set (as indicated by the inequality signs "less than" and not "less than or equal to").
• Shade the line segment between the open circles to show all numbers between \(-2\) and \(4\) are included in the solution set.

This graphical representation is intuitive and helps reinforce your understanding of the solution range.

Compound Inequalities

Compound inequalities involve two different inequalities combined by the word 'and' or 'or'. In the example given, the problem \-11 < -4x+5 < 13\ is a compound inequality connected by 'and'. This means we're finding values of \(x\) that make both parts of the inequality true simultaneously.

To solve a compound inequality like this, you must treat both parts of the inequality separately at first and then address them together:

• Start by isolating \(-4x\) as you do with any regular inequality, often working with each segment at a time, reducing all parts of the compound inequality together.
• In the first step, subtract 5 from all segments, simplifying the inequality.
• Once simplified to \-16 < -4x < 8\, solve for \(x\) as usual but remember to maintain the relationships between the different parts.

The goal is to find the range of \(x\) that satisfies both conditions, which results in a solution that can be neatly represented on a number line and interpreted easily.

Reversing Inequality Signs

Reversing inequality signs is a crucial step often encountered when solving inequalities. Specifically, when you divide or multiply all parts of an inequality by a negative number, you must "flip" the inequality sign to maintain the mathematical truth.

In our exercise, after simplifying the inequality to \-16 < -4x < 8\, the next step is solving for \(x\) by dividing by -4. This step necessitates reversing the inequality signs:

• Dividing each section by -4, we move from \-16 < -4x < 8\ to \4 > x > -2\.
• This reversal ensures the solution remains valid and reflects an important feature of inequalities.
  Without flipping the signs, the inequalities would incorrectly describe the solutions.

Always keep in mind: the direction of the inequality matters, and reversing signs is fundamental when negating or dividing inequalities. It is a simple but key principle that maintains the integrity of the solution.