Chapter 7: Problem 67
Solve the inequality. Then graph its solution. $$2 x-6<-7 \text { or } 2 x-6>5$$
Short Answer
Expert verified
The solution is \(x < -0.5\) or \(x > 5.5\).
Step by step solution
01
Solve first inequality
Start by solving the inequality \(2x - 6 < -7\). To do this, add 6 to both sides to isolate \(2x\), thus obtaining \(2x < -1\). Now, divide each side by 2 to solve for \(x\), getting \(x < -0.5\).
02
Solve second inequality
Similarly, to solve the inequality \(2x - 6 > 5\), add 6 to both sides to isolate \(2x\), resulting in \(2x > 11\). Now, divide both sides by 2 to get \(x > 5.5\).
03
Graph the solution
To graph the solution on a number line, start by marking the values of -0.5 and 5.5. Then, because \(x < -0.5\), draw a line to the left of -0.5 on the number line. Similarly, because \(x > 5.5\), draw a line to the right of 5.5 on the number line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
Visualizing solutions of inequalities can greatly enhance the understanding of what these solutions represent. Graphing can also be a powerful tool to check your work for errors. When graphing inequalities such as \(2x - 6 < -7\) or \(2x - 6 > 5\), the number line is your canvas. You typically start by marking critical values – in this example, \(-0.5\) and \(5.5\) – which are the boundary points where the inequality 'changes' from true to false or vice versa.
Remember, these points themselves are not part of the solution if the inequality symbol is '<' or '>', as opposed to '\(\right)\)' sign on the number line and draw an open circle to represent this point. Conversely, we use a closed dot if the inequality includes the value at the boundary (denoted by '\(\right)\)' or '\(\leq\)') indicates inclusions, which require a closed circle to denote that the endpoint is part of the solution.
For \(x < -0.5\), you would draw an arrow extending to the left from the open circle at \(-0.5\), indicating all real numbers less than \(-0.5\) are in the solution set. Similarly, an arrow extending to the right from the open circle at \(5.5\) would represent all the numbers greater than \(5.5\). This visualization solidifies your understanding and can be a handy reference for checking solutions to inequality problems.
Remember, these points themselves are not part of the solution if the inequality symbol is '<' or '>', as opposed to '\(\right)\)' sign on the number line and draw an open circle to represent this point. Conversely, we use a closed dot if the inequality includes the value at the boundary (denoted by '\(\right)\)' or '\(\leq\)') indicates inclusions, which require a closed circle to denote that the endpoint is part of the solution.
For \(x < -0.5\), you would draw an arrow extending to the left from the open circle at \(-0.5\), indicating all real numbers less than \(-0.5\) are in the solution set. Similarly, an arrow extending to the right from the open circle at \(5.5\) would represent all the numbers greater than \(5.5\). This visualization solidifies your understanding and can be a handy reference for checking solutions to inequality problems.
Inequality Solutions
Understanding solutions to inequalities is crucial since they often represent a range of possibilities, rather than a single answer as with equations. In the case of the inequalities \(2x - 6 < -7\) and \(2x - 6 > 5\), the solutions are \(x < -0.5\) and \(x > 5.5\), respectively. This specifies two distinct sets of numbers that satisfy the original compound inequality: all real numbers less than \(-0.5\) and all real numbers greater than \(5.5\).
These are infinite sets because between any two numbers on the number line lies another number. Hence, 'solutions to inequalities' often means identifying the general conditions that numbers must meet rather than finding a fixed number. While working with 'or' inequalities such as the ones given, the combined solution set is the union of the individual solution sets, meaning a number only needs to satisfy one (not necessarily both) of the inequalities to be considered a solution. When you're improving your understanding of inequality solutions, try using number line visualizations and test various numbers to see if they satisfy the conditions set by the inequality.
These are infinite sets because between any two numbers on the number line lies another number. Hence, 'solutions to inequalities' often means identifying the general conditions that numbers must meet rather than finding a fixed number. While working with 'or' inequalities such as the ones given, the combined solution set is the union of the individual solution sets, meaning a number only needs to satisfy one (not necessarily both) of the inequalities to be considered a solution. When you're improving your understanding of inequality solutions, try using number line visualizations and test various numbers to see if they satisfy the conditions set by the inequality.
Algebraic Inequalities
Algebraic inequalities are similar to equations, but instead of indicating equality, they show a relationship of inequality between two expressions. They are essential for expressing conditions and constraints in various fields, including engineering, economics, and sciences. Solving algebraic inequalities typically involves the same operations you would use for equations: addition, subtraction, multiplication, and division, while keeping in mind the special rule that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
In the step-by-step solution provided for \(2x - 6 < -7\) and \(2x - 6 > 5\), notice how we treated each part separately and performed arithmetic operations to isolate \(x\). Understanding that the solution to an inequality is not limited to one value but a range of values is a key concept in algebra that applies to many real-world scenarios. For example, when planning a budget, the amount you can spend (represented by an inequality) is not a fixed number but a range within your budget constraints. To practice solving algebraic inequalities, use systematic steps to isolate the variable and always consider the direction of the inequality sign, which provides crucial information on the solution set.
In the step-by-step solution provided for \(2x - 6 < -7\) and \(2x - 6 > 5\), notice how we treated each part separately and performed arithmetic operations to isolate \(x\). Understanding that the solution to an inequality is not limited to one value but a range of values is a key concept in algebra that applies to many real-world scenarios. For example, when planning a budget, the amount you can spend (represented by an inequality) is not a fixed number but a range within your budget constraints. To practice solving algebraic inequalities, use systematic steps to isolate the variable and always consider the direction of the inequality sign, which provides crucial information on the solution set.
