Solve the equations and inequalities. (2)/(5)<(2 x+3)/(5)-(3 x+1)/(7)<(6)/(7)

Name: Problem 68, Chapter 4: Understanding Elementary Algebra with Geometry
Uploaded: 2020-06-24T18:35:28-08:00
Duration: 2 min 40 s
Channel: Julie Silva
Description: Solve the equations and inequalities. (2)/(5)<(2 x+3)/(5)-(3 x+1)/(7)<(6)/(7)

step 1 We're being asked to solve the given compound inequality. So the first thing we need to do is ge

Solve the equations and inequalities. (2)/(5)<(2 x+3)/(5)-(3...

Key Concepts

Compound Inequalities

Compound inequalities involve finding solutions that satisfy more than one inequality simultaneously. The final solution to a compound inequality is the intersection of the solution sets for the individual inequalities, representing the range of values that make all parts of the compound inequality true.

Linear Inequalities

Linear inequalities involve expressions where variables appear in first degree and are compared using inequality symbols (such as <, >, ?, or ?). The solution to such inequalities is the set of values that makes the entire inequation true. Solving them entails applying algebraic operations similar to those used in linear equations, while also paying attention to the properties of inequalities, such as reversing the inequality sign when multiplying or dividing by a negative number.

Rational Expressions

Rational expressions consist of ratios where both the numerator and denominator are polynomials. In the context of inequalities, working with rational expressions typically requires combining or comparing fractions, which necessitates steps like finding a common denominator. Understanding the behavior and domain of these expressions is crucial because the values that make the denominator zero are excluded from the solution set.

Common Denominator

Finding a common denominator is a technique used to combine fractions that have different denominators. This process simplifies the manipulation of rational expressions by rewriting them with a shared denominator, thereby enabling the addition or subtraction of these fractions in a unified form.

Multiplication by the Least Common Multiple (LCM)

Multiplying every term in an equation or inequality by the least common multiple of the denominators is an effective method to clear the fractions from the expression. This step simplifies further algebraic manipulations, and in the context of inequalities, it is important to consider the sign of the LCM to ensure that the direction of the inequality is preserved.