Problem 111 - Algebra and Trigonometry Real Mathematics, Real People

Question

Without performing any calculations, match the inequality with its solution. Explain your reasoning. (a) \(2 x \leq-6\) (b) \(-2 x \leq 6\) (c) \(|x+2| \leq 6\) (d) \(|x+2| \geq 6\) (i) \(x \leq-8\) or \(x \geq 4\) (ii) \(x \geq-3\) (iii) \(-8 \leq x \leq 4\) (iv) \(x \leq-3\)

Step-by-Step Solution

Answer

(a) matches with (iv), (b) matches with (ii), (c) matches with (iii), (d) matches with (i).

Step 1: Match Inequality (a) with its Solution

Inequality (a): \(2x \leq -6\). This inequality can be simplified by dividing both sides by 2, which gives \(x \leq -3\). Hence, inequality (a) matches with Solution (iv).

Step 2: Match Inequality (b) with its Solution

Inequality (b): \(-2x \leq 6\). This inequality can be simplified by dividing both sides by -2, but it's important to remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So this simplifies to \(x \geq -3\), which matches with Solution (ii).

Step 3: Match Inequality (c) with its Solution

Inequality (c): \(|x + 2| \leq 6\). The absolute value of a number is always non-negative. When we have an inequality with an absolute value, it typically results in a range of values. This inequality says that the distance of \(x\) from -2 is less than or equal to 6, which means \(x\) is situated between -8 and 4, inclusive. So, inequality (c) matches with Solution (iii).

Step 4: Match Inequality (d) with its Solution

Inequality (d): \(|x + 2| \geq 6\). This means the distance of \(x\) from -2 is greater than or equal to 6. In other words, \(x\) is less than or equal to -8 or greater than or equal to 4. Hence, inequality (d) matches with Solution (i).

Key Concepts

Absolute Value Inequalities

When dealing with absolute value inequalities, it's important to remember that the absolute value of a number represents its distance from zero on a number line. This concept can be extended to describe the distance of a variable from a particular number. For example, when we have an inequality such as \(|x + 2| \leq 6\), it describes all values of \(x\) whose distance from -2 is no more than 6. This can be translated to the inequality

• -8 ≤ \(x\) ≤ 4

This range includes all numbers between -8 and 4, including the endpoints.

On the other hand, an inequality like \(|x + 2| \geq 6\) states that the distance from -2 is at least 6. Therefore, \(x\) can be any number less than or equal to -8 or greater than or equal to 4. Applying absolute value means there's a split in the range of possible solutions, specifically:

• \(x \leq -8\) or \(x \geq 4\)

Inequality Solutions

Understanding how to find solutions to inequalities is a foundational algebra skill. Like equations, inequalities can often involve operations such as addition, subtraction, multiplication, or division. The key difference is that when multiplying or dividing both sides by a negative number, the inequality sign reverses direction.

Take for instance the inequality \(\-2x \leq 6\). When dividing both sides by -2, remember to flip the inequality sign, resulting in:

• \(x \geq -3\)

This process ensures that the inequality remains true. Another example is the simple inequality \(2x \leq -6\). By dividing both sides by 2, we maintain the direction of the inequality, leading to:

• \(x \leq -3\)

Algebraic Inequalities

Algebraic inequalities encompass a broad spectrum of expressions where unknowns are compared using inequality signs such as \(>\), \(<\), \(\geq\), and \(\leq\). Solving these requires a systematic approach, performing operations that maintain the equivalence of the inequality while manipulating it to isolate the variable.

For example, with the inequality \(\-2x \leq 6\), by dividing both sides by -2, we switch the inequality to \(x \geq -3\). This highlights a fundamental principle in algebra: reverse the inequality upon dividing by negative numbers.

Moreover, algebraic inequalities can integrate absolute values, leading to solutions that span ranges rather than single points. Consider \(|x + 2| \leq 6\), where the solution implies a range of double inequalities:

• -8 ≤ \(x\) ≤ 4

Effective handling of inequalities involves clear understanding of properties and rules, ensuring accurate solution sets.