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Chapter 2: Problem 26

Solve each inequality and graph the solution on the number line. $$5 x+7 < 2 x+1$$

Short Answer

Expert verified
The solution to the inequality is \(x < -2\). On a number line, this is represented by an open circle at -2 and an arrow pointing to the left, indicating all numbers less than -2.

Step by step solution

01

Simplify the inequality

Begin by grouping like terms together: subtract \(2x\) from both sides to get \(3x + 7 < 1\).
02

Solve for x

Subtract 7 from both sides of the inequality to isolate \(x\) on the left side. This results in \(3x < -6\). To find \(x\), divide both sides by 3. So, \(x < -2\).
03

Graph the solution on the number line

Draw a number line and place a point at -2. Since \(x\) is less than -2, the solution would be any number that is to the left of -2. Thus, an open circle is placed on -2 and an arrow pointing left (towards smaller numbers) from -2 should be drawn

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Inequalities
When working with algebraic inequalities, we deal with mathematical expressions that show how one quantity is less than, greater than, or not equal to another. These are represented by symbols like < (less than), > (greater than), 鈮� (less than or equal to), and 鈮� (greater than or equal to).
In our exercise, we encounter the inequality \(5x + 7 < 2x + 1\). To solve this, our first goal is to isolate the variable, \(x\), on one side of the inequality. This process often requires using inverse operations to simplify and move terms around.
  • Subtract \(2x\) from both sides to eliminate the \(x\) from the right side of the inequality.
  • Subtract 7 from both sides to isolate terms without \(x\).
  • Finally, divide by the coefficient of \(x\) to solve for the variable.
Following these steps leads to a solution for \(x\), where we find the range of values that \(x\) can take to satisfy the original inequality.
Graphing Number Line
Graphing on a number line is a visual way to represent solutions to inequalities. It allows us to clearly see the set of all possible values that satisfy the inequality. Here are key steps for graphing solutions:
  • Firstly, draw a horizontal line and mark it into equal segments to represent numbers. Make sure to include the value that you have solved for in the inequality, in this case \(x < -2\).
  • Place an open circle on the number line at the value of \(x\) solved from the inequality to show that this value is not included in the solution set.
  • For a '<' or '鈮�' inequality, draw an arrow pointing left to show all numbers below that value are included. Conversely, for a '>' or '鈮�' inequality, an arrow would point right.
It's important to use an open circle for '<' and '>' inequalities, indicating that the exact number is not part of the solution set, and a closed circle for '鈮�' and '鈮�', indicating the number is included.
Inequality Solutions
Understanding inequality solutions means recognizing not just a single answer but a range or a set of numbers that fulfill the inequality's condition. To find the solution of an inequality:
  • As shown in the previous steps, simplify and solve the inequality algebraically first to identify critical values. For our example, \(x < -2\) signifies that any number smaller than -2 is part of the solution set.
  • Represent this solution graphically; assess whether the set includes boundary points or not. In our case, the boundary point -2 is not part of the solution, which is indicative of the inequality symbol '<', suggesting values less than and not including -2.
  • Finally, verify by substituting numbers from the solution set to ensure they satisfy the original inequality. In this example, if you choose a number less than -2, say -3, and substitute it back into the original inequality, the condition holds true, which confirms the solution.
These solutions can be applied to more complex inequalities including those with absolute values, variables on both sides, or those that require additional algebraic manipulation such as factoring.

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