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Chapter 2: Problem 97

Solve each inequality. Graph the solution set and write it in interval notation. $$ x^{2}+6 x-10

Short Answer

Expert verified
The solution is \(x < \frac{5}{8}\), and in interval notation, it is \((-\infty, \frac{5}{8})\).

Step by step solution

01

Simplify the Inequality

Start by simplifying both sides of the inequality. The original inequality is:\[x^2 + 6x - 10 < x(x - 10)\]Expand the right side:\[x^2 + 6x - 10 < x^2 - 10x\]To simplify further, subtract \(x^2\) from both sides to get:\[6x - 10 < -10x\]
02

Isolate the Variable

To isolate the variable \(x\), add \(10x\) to both sides:\[6x + 10x - 10 < -10x + 10x\]Simplify:\[16x - 10 < 0\]
03

Solve for x

Add 10 to both sides of the inequality:\[16x - 10 + 10 < 0 + 10\]This simplifies to:\[16x < 10\]Now, divide both sides by 16 to solve for \(x\):\[x < \frac{10}{16}\]Which simplifies to:\[x < \frac{5}{8}\]
04

Graph the Solution Set

On a number line, graph the inequality \(x < \frac{5}{8}\). Draw an open circle at \(\frac{5}{8}\) and shade the line to the left to indicate that \(x\) can be any number less than \(\frac{5}{8}\).
05

Write in Interval Notation

The solution set in interval notation is:\[(-\infty, \frac{5}{8})\]

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Key Concepts

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

Graphing Inequalities
Graphing inequalities involves showing on a number line all the possible values that satisfy an inequality expression. In this exercise, we found that the inequality solution was \(x < \frac{5}{8}\). To accurately represent this on a number line:
  • First, locate the point \(\frac{5}{8}\) on the number line.
  • Next, draw an open circle at \(\frac{5}{8}\). This symbolizes that \(\frac{5}{8}\) is not included in the solution set.
  • Finally, shade the portion of the number line to the left of \(\frac{5}{8}\). This shaded area represents all the numbers that are less than \(\frac{5}{8}\).
Graphing is a visual method to confirm and communicate the range of potential solutions effectively. By using open or closed circles (for non-strict or strict inequalities), you convey whether endpoints are included or excluded from the solutions.
Interval Notation
Interval notation is a concise way to describe a range of numbers that represent the solution to an inequality. In our inequality solution \(x < \frac{5}{8}\), interval notation is used to express all numbers less than \(\frac{5}{8}\). The interval notation for this solution is \((-abla, \frac{5}{8})\):
  • The round parenthesis \((\) or \()\) indicates that \(\frac{5}{8}\) itself is not included in the interval, aligning with the open circle concept from graphing.
  • The symbol \(-\infty\) represents all numbers going forever in the negative direction.
This format is efficient for writing and interpreting inequalities, as it succinctly describes the set of numbers involved without listing them individually.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations combined to express a particular value or relationship. Simplifying these expressions is crucial for solving inequalities as seen in our exercise. The original inequality was:\[x^2 + 6x - 10 < x(x - 10)\]Here's a breakdown of handling algebraic expressions in this inequality:
  • First, expand expressions to eliminate parentheses, producing equations that are easier to compare. For example, the right side expanded to \(x^2 - 10x\).
  • Next, simplify by removing like terms or combining them. Here, subtracting \(x^2\) from both sides removed it entirely.
  • Finally, isolate the variable by performing arithmetic operations on both sides of the inequality. Our goal was \(x < \frac{5}{8}\), achieved by adding/subtracting terms and dividing the equation by a constant factor.
Simplification transforms complex problems into straightforward ones, revealing the solution set quickly.

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