Find all values of x that satisfy both inequalities simultaneously. (a) 3 x+7>1 and 2 x+1<3 (b)

step 1 To find all values of x that can satisfy the inequality, we have 3x plus 7 equal to 1. And you h

Find all values of x that satisfy both inequalities...

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Key Concepts

Interval Intersection

This concept involves determining the common values shared between two or more solution sets. In practice, one solves each inequality separately and then identifies the overlapping interval, which represents the final answer.

Simultaneous Inequalities

When dealing with more than one inequality at a time, you must find the intersection of their solution sets, meaning only the values that satisfy every inequality are valid solutions. This concept is fundamental when constraints are applied concurrently in problems.

Linear Inequalities

These are inequalities involving a linear expression. The process of solving them usually involves isolating the variable on one side through operations such as addition, subtraction, multiplication, or division. An essential part of this process is remembering to reverse the inequality sign if you multiply or divide by a negative number.

Solution Sets

The result of solving a linear inequality is a set of numbers that satisfy the inequality. These sets are often represented in interval notation and visually depicted on a number line, allowing for a clear understanding of the range of possible solutions.

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