Question

Show that the indicated implication is true $$ |x+2|<0.3 \Rightarrow|4 x+8|<1.2 $$

Short answer

The implication is true since \(-2.3 < x < -1.7\) ensures \(|4x+8|<1.2\) when \(|x+2|<0.3\).

Step-by-step solution

Understanding the Problem

We need to prove that if \(|x+2|<0.3\), then \(|4x+8|<1.2\) is true. This is a direct implication problem that requires manipulation and substitution of inequalities.

Analyzing the Given Inequality

The initial inequality is \(|x+2|<0.3\). This tells us that the value of \(x\) is at most 0.3 units away from -2. We can rewrite this inequality as \[-0.3 < x+2 < 0.3\].

Solving for x

We need to solve the double inequality for \(x\). By subtracting 2 from all parts of the inequality, we have \[-0.3 - 2 < x < 0.3 - 2\], which simplifies to \[-2.3 < x < -1.7\].

Transforming the Goal Inequality

The goal is to show that \(|4x+8| < 1.2\). Notice that \(4x+8 = 4(x+2)\). Therefore, we can substitute this into the absolute value inequality to obtain \(|4(x+2)| < 1.2\).

Substituting the x+2 Boundaries

Since we have the inequality \(-0.3 < x+2 < 0.3\), multiplying all parts by 4, we get \(-1.2 < 4(x+2) < 1.2\).

Verifying the Goal

The inequality \(-1.2 < 4(x+2) < 1.2\) can be rewritten as \(|4(x+2)|<1.2\). This matches exactly with the inequality we needed to prove, \(|4x+8|<1.2\). Thus, the implication is true.

Key concepts

Absolute Value Inequalities

An absolute value inequality involves the absolute value of an expression and typically states how far the expression can "stretch" from zero. The absolute value of a real number is its distance from zero on the number line, without considering direction. When we encounter inequalities involving absolute values, they usually have a form like \( |expression| < some ext{ }number \) or \( |expression| > some ext{ }number \).

For instance, in the exercise, \( |x+2| < 0.3 \) implies that the variable \( x+2 \) is within 0.3 units of zero. This can be split into a double inequality: \(-0.3 < x+2 < 0.3\). This means if you add 2 to x, the result must still fall within the range from \(-0.3\) to \(+0.3\). Understanding this property is key to solving and manipulating absolute value inequalities, which is a fundamental concept in algebra and calculus.

Direct Implication Problems

A direct implication problem requires proving that a certain condition, if fulfilled, leads directly and inevitably to another condition being true. In the context of the problem, we start by understanding that if \( |x+2| < 0.3 \), then \( |4x+8| < 1.2 \) must also be true.

Such problems are approached by logically maneuvering through the inequalities, often starting from the initial condition and logically progressing to the conclusion using properties of equalities and inequalities. Here, the direct implication stems from recognizing that the inequality for \( 4x+8 \) can be transformed using both algebraic manipulation and by substituting equivalent expressions. This guides us to the solution by systematically demonstrating that the given initial inequality directly satisfies the condition needed for the final result.

Double Inequality Solution

A double inequality is one where there are two inequalities concerning a single variable. This type often arises from solving absolute value equations. For example, in the given problem, when \( |x+2| < 0.3 \) is expanded into \(-0.3 < x+2 < 0.3\), it creates a double inequality.

The steps to solve a double inequality involve making the variable 'stand-alone' by isolating it. For \( -0.3 < x+2 < 0.3 \), we subtract 2 from all three parts of the inequality to solve for x, resulting in \(-2.3 < x < -1.7\). This solution provides the range of x that satisfies the original absolute value inequality.

• Ensure that operations are applied across all parts of the inequality.
• Keep the inequality balanced to maintain correctness.

Handling double inequalities demands careful attention to detail in order to preserve the inequalities' directions and the logical consistency of the solution.

Mathematical Proof Techniques

Proof techniques in mathematics allow us to validate statements and implications. In this problem, applying these techniques means demonstrating that given the inequality \( |x+2| < 0.3 \), the implication naturally follows that \( |4x+8| < 1.2 \).

The approach usually begins with recognizing simplifications or substitutions that can bridge the initial and final conditions. For instance, the simplification \( 4x+8 = 4(x+2) \) helps illustrate why and how the transformation from \( |x+2| < 0.3 \) to \( |4x+8| < 1.2 \) holds true.

• One must show all logical steps comprehensively.
• Each manipulation must be justifiable either through known axioms or previously proven statements.
• Substitution is vital to substituting complex parts with simpler expressions.

Solid proof techniques are essential in mathematics, not merely to "get the right answer," but to ensure understanding and correctness of the implication or statement.