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Magazine Chapter 1: Problem 24
Graph all solutions on a number line and provide the corresponding interval notation. ?12+5(5?2x)<83
Short Answer
Expert verified
Graph: Hollow circle at -4.6, shade right. Interval: \((-4.6, \,
ightarrow)\).
Step by step solution
01
Distribute the 5
The inequality given is \(12 + 5(5 - 2x) < 83\). Start by distributing the 5 into the terms inside the parenthesis. This gives: \(12 + 25 - 10x < 83\).
02
Simplify the Equation
Combine the constant terms on the left side of the inequality: \(12 + 25 = 37\). So the inequality now is \(37 - 10x < 83\).
03
Move Constants to the Right Side
Subtract 37 from both sides of the inequality to isolate the term with the variable. This results in \(-10x < 46\).
04
Solve for x
Divide each side of the inequality by -10, remembering to flip the inequality sign: \(x > -4.6\).
05
Graph the Solution
On a number line, mark a hollow circle at \(-4.6\) to indicate that it is not included (since the inequality is strictly greater than). Shade the region to the right of \(-4.6)\) to show all numbers greater than \(-4.6\).
06
Write the Interval Notation
Since \(x > -4.6\), the interval notation is \((-4.6, \,
ightarrow)\). This indicates all numbers greater than \(-4.6\), extending to positive infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Number Line
A number line is a visual representation of numbers in a linear format. It shows numbers in order along a one-dimensional line, typically horizontal, with each point corresponding to a number. When dealing with inequalities, the number line becomes particularly useful. It allows you to easily display possible solutions at a glance. To graph an inequality like the one from the exercise, follow these steps:
- Identify critical points: These are numbers where the inequality could change direction, such as solutions found from solving the inequality.
- Mark intervals: Indicate these critical numbers on the number line. If the inequality excludes the number (e.g., > or <), use an open circle. If the inequality includes the number (e.g., 鈮� or 鈮�), use a closed circle.
- Shade the solution: Highlight or shade the region where the inequality holds true. For our exercise, shading to the right of -4.6 represents all numbers greater than -4.6.
Interval Notation
Interval notation is a shorthand used to denote ranges of numbers and solutions to inequalities. It provides a concise way to convey which numbers are included in a set. Understanding interval notation boils down to knowing a few basic symbols:
- Round brackets ( ): These exclude an endpoint, used for strict inequalities such as > and <.
- Square brackets [ ]: These include an endpoint, used for inclusive inequalities such as 鈮� and 鈮�.
- Infinity (鈭�): Used to signify that the solution continues indefinitely. It is always paired with a round bracket since infinity isn鈥檛 a number that's included in the set.
- All numbers greater than -4.6 are part of the solution, but -4.6 itself is not included, as indicated by the round bracket.
- The arrow towards infinity indicates that there is no upper bound to the solution; it stretches on without limit.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols (like +, -, *, /) that represent a particular value or set of values. In solving inequalities, manipulating algebraic expressions is crucial:
- Distributive Property: Used to expand expressions, such as distributing a multiplier like the +5 into a parenthesis as shown in the exercise: from \(5(5 - 2x)\) to \(25 - 10x\).
- Combining Like Terms: Simplify expressions by adding or subtracting terms that share the same variable, like simplifying \(12 + 25\) to \(37\).
- Isolation of Variables: Rearranging the inequality to isolate the variable on one side, as in \(-10x < 46\).
- Dividing by Negative Numbers: Remember that dividing (or multiplying) both sides of an inequality by a negative number reverses the inequality sign. This is seen when dividing by -10 to solve for x.
