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Chapter 5: Problem 34

Solve each inequality algebraically. $$(x-3)(x+2)

Short Answer

Expert verified
The solution is \( x > -\frac{11}{4} \).

Step by step solution

01

Expand both sides of the inequality

First expand the left-hand side of the inequality: \( (x-3)(x+2) \rightarrow x^2 - x - 6 \)
02

Set up the inequality with expanded terms

Rewrite the original inequality with the expanded terms: \( x^2 - x - 6 < x^2 + 3x + 5 \)
03

Simplify the inequality

Subtract \( x^2 \) from both sides: \( -x - 6 < 3x + 5 \)
04

Collect all x terms on one side

Add \( x \) to both sides to isolate the variable terms on one side: \( -6 < 4x + 5 \)
05

Isolate the term with x

Subtract 5 from both sides to isolate the term with x: \( -6 - 5 < 4x \) Simplify: \( -11 < 4x \)
06

Solve for x

Divide both sides by 4: \( \frac{-11}{4} < x \) Express the solution in a simpler form: \( x > -\frac{11}{4} \)

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Key Concepts

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

Solving Inequalities
Inequalities are similar to equations, but instead of an equals sign (=), they use inequality signs like <, >, 鈮�, and 鈮�. Solving inequalities means finding the range of values for the variable that makes the inequality true. To do this systematically, we follow a set of algebraic steps to simplify and isolate the variable, much like solving equations. However, keep in mind that when you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed.
Expanding Expressions
Expanding expressions involves distributing and simplifying terms to rewrite a mathematical statement in a different form. In our problem, we start by expanding \((x-3)(x+2)\). To expand this, use the distributive property (FOIL method for binomials):

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Most popular questions from this chapter

Find \(g(3)\) where \(g(x)=\left\\{\begin{array}{ll}3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0\end{array}\right.\) Find \(g(3)\) where $$ g(x)=\left\\{\begin{array}{ll} 3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0 \end{array}\right. $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=3 x^{4}-3 x^{3}-5 x^{2}+27 x-36 $$

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