Problem 50 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. $$ -2<-3 x+1<10 $$

Step-by-Step Solution

Answer

The solution set to the inequality is \( -3 < x < 1 \), which means \( x \) can be any number between \( -3 \) and \( 1 \), but not including \( -3 \) or \( 1 \). This is represented on a number line with a line from \( -3 \) to \( 1 \) with open circles at the ends.

Step 1: Removing the Constant on the Rightmost Inequality

Subtract \( 1 \) from all three sides of the inequality to get the equivalent inequality \( -3 < -3x < 9 \).

Step 2: Isolating x

After removing the constant, divide each part of the inequality by \( -3 \) (which is a negative number). Now remember, whenever you multiply or divide both sides of an inequality by a negative number, you should flip the direction of the inequality. So, we will get \( 1 > x > -3 \).

Step 3: Reordering the Inequality

To make the inequality more intuitive, reverse it, yielding \( -3 < x < 1 \). We can read this as \( x \) is greater than \( -3 \) and less than \( 1 \).

Step 4: Graphing the Solution

On a number line, mark points at \( -3 \) and \( 1 \). A parenthesis or open circle is used to denote \( -3 \) and \( 1 \) because the inequality symbol does not include equality. Draw a line between \( -3 \) and \( 1 \) to indicate all the numbers \( x \) could be.

Key Concepts

Solving Compound Inequalities

Solving compound inequalities can seem a little tricky at first, but with practice, it becomes much easier! Essentially, a compound inequality is like solving two inequalities at once. Consider the inequality \(-2 < -3x + 1 < 10\). To solve the inequality, follow these basic steps:

• Start by isolating the variable term. For this, you need to remove any constants from both sides of the inequality.
• Next, deal with the variable term separately in both parts of the compound inequality. In our example, subtract 1 from each part of the inequality to eliminate the constant.
• Don’t forget about flipping the inequality direction if you multiply or divide by a negative number!
• Once you isolate the variable, you should rewrite it in an intuitive order.

The key here is to follow systematic steps and treat each part of the compound inequality separately until you're left with a clean inequality that is easy to interpret.

Graphing Inequalities

After solving the inequality, the next step is to represent the solution graphically.

• Use a number line, which is an effective tool for visualizing solutions to inequalities.
• Mark the critical points identified in the inequality. For example, in \(-3 < x < 1\), the critical points are \(-3\) and \(+1\).
• Since the inequality does not include equals signs, use open circles to denote \(-3\) and \(+1\). This tells us these points are not included in the solution.
• Draw a line or a segment between these points to indicate the range of values that satisfy the inequality. This line represents all the values that \(x\) can take to make the original inequality true.

Graphing inequalities not only helps in visualizing the range of solutions but also reinforces the understanding of the inequality structure and its solutions.

Inequality Properties and Rules

Understanding the properties and rules that govern inequalities is essential for solving them correctly. Here are some key rules:

• Inequality Transposition: You can add or subtract the same value from all parts of an inequality without changing its direction.
• Multiplication and Division: If you multiply or divide all parts of an inequality by a positive number, the inequality direction remains unchanged. However, if you use a negative number, the inequality sign must be flipped. This is crucial to remember, as seen in our example where we divide by \(-3\).
• Order of Inequalities: Inequalities can usually be rewritten in a way that makes comparison easier, such as rewriting \(1 > x > -3\) as \(-3 < x < 1\) for better readability.

These rules form the foundation of solving and understanding inequalities. By applying these principles, you can confidently tackle any inequality problem.