Problem 40 - Elementary Algebra

Question

Graph the compound inequality \(-6 \leq-2 x+2<-4\).

Step-by-Step Solution

Answer

Question: Graph the compound inequality \(-6 \leq -2x + 2 < -4\) and identify its solution set. Answer: The solution set of the compound inequality \(-6 \leq -2x + 2 < -4\) is \(3 < x \leq 4\).

Step 1: Solve the first inequality

Solve the inequality \(-6 \leq -2x + 2\). To do this, follow these steps: 1. Subtract 2 from both sides: \(-8 \leq -2x\) 2. Divide by -2 (and remember to flip the inequality sign when dividing by a negative number): \(4 \geq x\) So, our first inequality is \(x \leq 4\).

Step 2: Solve the second inequality

Solve the inequality \(-2x + 2 < -4\). To do this, follow these steps: 1. Subtract 2 from both sides: \(-2x < -6\) 2. Divide by -2 (and remember to flip the inequality sign when dividing by a negative number): \(x > 3\) So, our second inequality is \(x > 3\).

Step 3: Find the intersection of the solution sets

To determine the intersection of the solution sets for \(x \leq 4\) and \(x > 3\), we look for the values of x that satisfy both conditions. Since \(3 < x \leq 4\), we can write the compound inequality as \(3 < x \leq 4\).

Step 4: Graph the compound inequality on a number line

Create a number line, and mark the critical points at x = 3 and x = 4. 1. For x = 3, use an open circle because the inequality is \(x > 3\), not \(x \geq 3\). 2. For x = 4, use a closed circle because the inequality is \(x \leq 4\), not \(x < 4\). 3. Draw a line connecting the two circles, indicating all the values between 3 and 4 that satisfy the compound inequality. The graph represents the solution set \(3 < x \leq 4\) for the compound inequality \(-6 \leq -2x + 2 < -4\).

Key Concepts

Solving Inequalities

Inequalities are similar to equations but instead of an equal sign, they use inequality symbols like \(<, >, \leq, \geq\). Solving an inequality involves finding the values of the variable that make the inequality true. The steps involve simplifying each part separately.
To solve \(-6 \leq -2x + 2\), first subtract 2 from both sides to get \(-8 \leq -2x\). The next step is dividing both sides by -2, which flips the inequality sign, resulting in \(4 \geq x\). This gives us \(x \leq 4\), meaning x can be 4 or any number less than 4.
The second inequality, \(-2x + 2 < -4\), is simplified similarly. Subtract 2 from both sides: \(-2x < -6\) and divide by -2 to get \(x > 3\). Remember, dividing by a negative number flips the inequality. These steps show how to handle inequalities efficiently.

Number Line Graph

A number line is a visual tool to represent solution sets of inequalities. This helps to understand which numbers satisfy a compound inequality.
For the inequality \(3 < x \leq 4\), the number line should mark critical points at x = 3 and x = 4.

• Use an open circle at x = 3 because the inequality is "greater than," excluding 3.
• Use a closed circle at x = 4 because the inequality includes 4.

Draw a line between these points, which illustrates the range of x values that satisfy the compound inequality. The number line clearly shows all numbers greater than 3 and less than or equal to 4.

Intersection of Solution Sets

When dealing with compound inequalities, you often look for the intersection. The intersection is where two or more sets of solutions overlap.
For the two inequalities, \(x \leq 4\) and \(x > 3\), you want to find where both conditions are true at the same time. Drawing them on a number line separately, you combine the regions where they overlap.
The overlap for \(x \leq 4\) and \(x > 3\) is \(3 < x \leq 4\). This means the solution to the compound inequality includes all x values that satisfy both conditions simultaneously.

Inverse Operations in Inequalities

Inverse operations are key when rearranging inequalities, especially when isolating variables. They involve doing the opposite operation to both sides of the inequality, like subtraction instead of addition or the reverse.
In our exercise, subtracting 2 from both sides of the inequalities brings the constant term to one side. Then, dividing by -2 isolates x, but it’s crucial to remember dividing or multiplying by a negative flips the inequality sign.
Without flipping, our solution could be incorrect, which highlights the importance of understanding how these operations affect inequalities. This principle is consistent and essential when solving any inequality.