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Chapter 8: Problem 90

Solve and graph each solution set. $$ 3 a-7>-10 \text { or } 5 a+2 \leq 22 $$

Short Answer

Expert verified
The solution set is \(a > -1\) or \(a \leq 4\).

Step by step solution

01

- Solve the first inequality

Start by solving the first inequality, \(3a - 7 > -10\). Add 7 to both sides:\[3a - 7 + 7 > -10 + 7\]This simplifies to:\[3a > -3\]Next, divide both sides by 3:\[a > -1\]
02

- Solve the second inequality

Now solve the second inequality, \(5a + 2 \leq 22\). Subtract 2 from both sides:\[5a + 2 - 2 \leq 22 - 2\]This simplifies to:\[5a \leq 20\]Next, divide both sides by 5:\[a \leq 4\]
03

- Combine the solution sets

The combined solution set is the union of both solutions: \[a > -1 \text{ or } a \leq 4\]This means any value of \(a\) greater than -1 or less than or equal to 4 satisfies the inequality.
04

- Graph the solution set

Draw a number line. Mark an open circle at -1 indicating that -1 is not included in the solution set (\(a > -1\)). Draw an arrow to the right from -1 to show all numbers greater than -1. Then, mark a closed circle at 4 indicating that 4 is included (\(a \leq 4\)), and draw a line extending leftwards to show all numbers less than or equal to 4. The union of these two will show the final solution on the graph.

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Key Concepts

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

Compound Inequalities
A compound inequality involves two or more inequalities linked together by the words 'and' or 'or'. There are two types of compound inequalities:
  • Conjunctions ('and'): Both inequalities must be true at the same time.
  • Disjunctions ('or'): At least one of the inequalities must be true.
In our example, the compound inequality is \(3a - 7 > -10 \text{ or } 5a + 2 \leq 22\), indicating a disjunction. This means the solution can satisfy either inequality or both.
Solution Sets
A solution set includes all the values that satisfy the given inequality or equations. For compound inequalities, the solution set combines all values that satisfy at least one of the inequalities when using 'or', or both when using 'and'.
In the given problem, solving each inequality separately produces:
  • For \(3a - 7 > -10\), we find that \(a > -1\).
  • For \(5a + 2 \leq 22\), we get \(a \leq 4\).
Since the inequalities are linked by 'or', the solution set includes all values greater than -1 and all values less than or equal to 4. This set can be written as \(a > -1 \text{ or } a \leq 4 \).
Graphing Inequalities
Graphing inequalities helps visualize the solution set on a number line. Here鈥檚 how to graph the solution for our example:
  • Start with a number line.
  • For \(a > -1\), place an open circle at -1 to indicate -1 is not part of the solution, and draw an arrow to the right to show all numbers greater than -1.
  • For \(a \leq 4\), place a closed circle at 4 to show 4 is included in the solution, and draw a line extending leftwards to show all numbers less than or equal to 4.
  • Combine the two parts to form the final graph. The number line will have an open circle at -1 with an arrow pointing right, and a closed circle at 4 with a line pointing left, indicating the union of both solutions.
This visual representation makes it easier to understand all possible values of \(a\) in the solution set.

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Most popular questions from this chapter

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