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Chapter 8: Problem 73

Solve and graph each solution set. $$ -7 \leq 2 a-3 \text { or } 3 a+1>7 $$

Short Answer

Expert verified
The solution set is \(a \leq -2 \text{ or } a > 2\).

Step by step solution

01

Break Down the Inequality

We have two inequalities to solve: 1. \(-7 \leq 2a - 3\)2. \(3a + 1 > 7\)
02

Solve the First Inequality

Add 3 to both sides of the inequality \(-7 \leq 2a - 3\) to isolate the term with \(a\):\(-7 + 3 \leq 2a\)Which simplifies to:\(-4 \leq 2a\)Now, divide both sides by 2:\(-2 \leq a\)
03

Solve the Second Inequality

Subtract 1 from both sides of the inequality \(3a + 1 > 7 \):\(3a > 6\)Now, divide both sides by 3:\(a > 2\)
04

Combine the Solution Sets

The solution to the compound inequality is the union of the two sets:1. \(-2 \leq a\)2. \(a > 2\)The combined solution set is \(a \leq -2 \text{ or } a > 2\).
05

Graph the Solution Set

Graph the solutions on a number line:1. For \(-2 \leq a\), draw a closed circle at \(-2\) and shade to the right.2. For \(a > 2\), draw an open circle at \(2\) and shade to the right.

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Key Concepts

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

Solving Inequalities
Inequalities in algebra are like equations but with a twist. Instead of showing equality, they show that one side is greater or smaller than the other. They use signs like <, >, 鈮�, and 鈮�. Here, inequalities show us a range of values instead of a single value.

To solve an inequality, you perform operations just like you would for an equation. Add, subtract, multiply, or divide both sides by the same number. Remember:
  • If you multiply or divide by a negative number, flip the inequality sign.
  • Keep the variable on one side for clarity.
As part of solving \(-7 \leq 2a - 3\) and \(3a + 1 > 7\), notice how we isolated the variable just like in regular algebra. Addition, subtraction, and division were used to get to \(-2 \leq a\) and \(a > 2\), respectively.
Compound Inequalities
Compound inequalities are a combination of two or more inequalities joined by 'and' or 'or'. In compound inequalities:
  • 'And' means both conditions must be true simultaneously.
  • 'Or' means at least one condition must be true.
For instance, the problem involved \(-7 \leq 2a - 3\) or \(3a + 1 > 7\). This means any value of \(a\) that satisfies either inequality will be part of the solution. Here, the result \(a \leq -2\) or \(a > 2\) means that both conditions do not have to hold at the same time.

In practice:
1. Solve each inequality separately.
2. Combine the results to find the overall solution set.
Graphing Solutions
Visualizing inequalities on a number line helps see where solutions lie. Here's how to graph:
  • Use a closed circle for \( \leq \text{ or } \geq \). This shows that the number is included in the solution.
  • Use an open circle for \( < \text{ or } \> \). This shows that the number is not included.
  • Shade the line to show where all numbers in the solution set lie.
For \(-2 \leq a\), place a closed circle on \(-2\) and shade to the right, indicating all numbers from \(-2\) and above are solutions. For \(a > 2\), place an open circle on \2\ and shade to the right, indicating numbers greater than \2\ are solutions. Combining these, the number line will show a closed circle on \(-2\) and shading to the right, along with an open circle at \2\ and shading to the right.
This visual aid makes it easier to understand and verify the solution set.

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