Problem 54
Question
Solve the inequality. Then graph the solution. \(8>2 x>-4\)
Step-by-Step Solution
Verified Answer
The solution to the inequality \(8>2x>-4\) is \(-2
1Step 1: Isolate the Variable
In order to solve the inequality, the variable needs to be isolated. To do this, all three sides of the inequality should be divided by 2.Therefore, the inequality becomes \(8/2 > 2x/2 > -4/2\).
2Step 2: Simplify the Inequality
Simplify the inequality to isolate x by working out the math on all three sides. This gives us \(4>x>-2\).
3Step 3: Interpret the Inequality
This inequality can be interpreted as x is greater than -2 AND less than 4.
4Step 4: Draw the Solution on a Number Line
To graph this solution on a number line, start by drawing a number line, then mark points corresponding to -2 and 4 on it. Since the inequality is 'less than' and 'greater than' (not 'less than or equal to' or 'greater than or equal to'), open circles should be used at -2 and 4 on the number line. Then shade the line between -2 and 4 to represent all possible values for x.
Key Concepts
Compound InequalitiesInequalities GraphingNumber Line Representation
Compound Inequalities
Compound inequalities are mathematical expressions that involve two separate inequalities joined together by the words 'and' or 'or'. In the context of this exercise, we have a compound inequality that is connected by 'and', expressed as \( 8>2x>-4 \).
This means that the solution to our compound inequality must satisfy both conditions at the same time. In other words, the values of \( x \) that make this true should be greater than \(-2\) and less than \(4\).
To solve it, we approach it step by step, treating it as one connected inequality. This is known as a 'conjunction', and finding the solution involves operations that treat each inequality part equally. By manipulating the inequalities as a unit, our goal is to isolate the variable, making it easier to understand the span of solutions that fit both parts of the inequality.
This means that the solution to our compound inequality must satisfy both conditions at the same time. In other words, the values of \( x \) that make this true should be greater than \(-2\) and less than \(4\).
To solve it, we approach it step by step, treating it as one connected inequality. This is known as a 'conjunction', and finding the solution involves operations that treat each inequality part equally. By manipulating the inequalities as a unit, our goal is to isolate the variable, making it easier to understand the span of solutions that fit both parts of the inequality.
Inequalities Graphing
Graphing inequalities is a wonderful visual way to represent solutions. Once you solve an inequality, graphing it can help you see the range of possible solutions at a glance. To graph an inequality, you typically use a number line.
For this exercise, once we solved the inequality \( 4>x>-2 \), the next step is to graphically represent this range. It's important to plot the solutions correctly, as visual representations provide quick understanding and verification of solutions.
For this exercise, once we solved the inequality \( 4>x>-2 \), the next step is to graphically represent this range. It's important to plot the solutions correctly, as visual representations provide quick understanding and verification of solutions.
- Locate the important boundary points, which are \(-2\) and \(4\) in this case.
- Open circles on the number line indicate that the boundary numbers themselves are not included in the solution. Closed circles would mean the number is part of the solution.
- Shade or draw a line between these two points to show all numbers that are greater than \(-2\) and less than \(4\).
Number Line Representation
A number line is a helpful tool for visual learners to understand and solve inequalities. It provides a straightforward visual representation of possible values for a given variable. When drawing a number line, begin by marking all the key numbers in the inequality.
In our example, the solutions span from \(-2\) to \(4\), so these values are marked as key reference points on the number line. Use an open circle to clearly indicate that neither value is included in the solution set. This is a crucial element, as using a solid or closed circle would imply inclusion.
After marking the boundaries, draw a line or shade the region between these open circles. This visually represents all the values that satisfy the compound inequality, providing a clear image of the solution set.
In our example, the solutions span from \(-2\) to \(4\), so these values are marked as key reference points on the number line. Use an open circle to clearly indicate that neither value is included in the solution set. This is a crucial element, as using a solid or closed circle would imply inclusion.
After marking the boundaries, draw a line or shade the region between these open circles. This visually represents all the values that satisfy the compound inequality, providing a clear image of the solution set.
- Start with a horizontal line as the number line.
- Mark and label numbers distinctly at \(-2\) and \(4\).
- Ensure open circles are drawn at these numbers to show that they are not included.
- Shade the line between the circles for easy visualization.
Other exercises in this chapter
Problem 54
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