Problem 52

Question

Solve each group of equations and inequalities analytically. (a) $|x-3|=5$ (b) $|x-3|>5$ (c) $|x-3|<5$

Step-by-Step Solution

Verified

Answer

(a) $x = 8$ or $x = -2$; (b) $x > 8$ or $x < -2$; (c) $-2 < x < 8$.

1Step 1: Solve the Equation |x-3|=5

For the equation $|x-3|=5$, we need to consider both possible cases for the absolute value: 1. $x-3 = 5$2. $x-3 = -5$Solving these:- For $x-3 = 5$, add 3 to both sides: $x = 8$.- For $x-3 = -5$, add 3 to both sides: $x = -2$.Thus, $x = 8$ or $x = -2$.

2Step 2: Solve the Inequality |x-3|>5

For the inequality $|x-3|>5$, consider two separate inequalities:1. $x-3 > 5$2. $x-3 < -5$Solving these:- For $x-3 > 5$, add 3 to both sides: $x > 8$.- For $x-3 < -5$, add 3 to both sides: $x < -2$.Thus, the solution is $x > 8$ or $x < -2$.

3Step 3: Solve the Inequality |x-3|<5

For the inequality $|x-3|<5$, consider the combined inequality: $-5 < x-3 < 5$.Solving this inequality:- Add 3 to all parts: $-5 + 3 < x < 5 + 3$, resulting in $-2 < x < 8$.Thus, the solution is $-2 < x < 8$.

Key Concepts

Solving InequalitiesAnalytical MethodsCombined Inequality Solutions

Solving Inequalities

When dealing with inequalities involving absolute values, it helps to break them down into more manageable parts. Absolute value inequalities express distances from zero, translating into two scenarios since the expression within the absolute value can be either positive or negative. Consider the inequality $|x-3|>5$. This means that the distance between $x$ and 3 is greater than 5. The key to solving such inequalities is to split them into two separate inequalities:

$x-3 > 5$
$x-3 < -5$

By solving these inequalities, you can determine that the solution is $x > 8$ or $x < -2$, illustrating that $x$ lies beyond 5 units from 3, either to the left or right on the number line.
In contrast, solving the inequality $|x-3|<5$ implies the distance between $x$ and 3 is less than 5, so we consider the combined inequality \(-5

Analytical Methods

Analytical methods involve breaking down problems into more straightforward mathematical expressions. This approach is particularly useful in solving absolute value equations or inequalities since it transforms a seemingly complex problem into a simpler one. For a problem like $|x-3|=5$, we interpret the absolute value as two potential equations. We know that $x-3$ could either equal 5 or -5:

$x-3 = 5$
$x-3 = -5$

By solving each equation separately, you quickly find that $x = 8$ or $x = -2$.
These precise calculations allow students to leverage known methods of solving linear equations, but in a context that includes absolute values. It simplifies the work required and results in accurate solutions for each scenario. Analytical thinking allows the breakdown of each component, ensuring nothing is overlooked.

Combined Inequality Solutions

With combined inequalities, particularly those involving absolute values, it’s essential to rewrite the inequality in terms of a continuous range. Consider the inequality $|x-3|<5$, which translates into the combined inequality \-5This indicates that $x$ lies within a range that extends 5 units in both directions from 3. To solve such inequalities, isolate $x$ by applying algebraic operations equally to all parts of the inequality:

Start with \(-5
Add 3 throughout: \(-5+3
Simplify to get \(-2

This method ensures that solutions reflect not just isolated values but a complete interval on the number line. The result, a number line representation, helps visualize the possible values of $x$, promoting a deeper understanding of the solution set.

Problem 52

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