Problem 10 - Elementary and Intermediate Algebra

Question

Use a check to determine whether \(-3\) is a solution of the -double linear inequality. a. \(-1<-3 x+4<12\) b. \(-1<-3 x+4<14\)

Step-by-Step Solution

Answer

Only part (b), \(-1 < -3x + 4 < 14\), has \(x = -3\) as a solution.

Step 1: Understanding the Double Inequality

A double inequality 3 < -3x + 4 < 12 consists of two separate inequalities: 3 < -3x + 4 and -3x + 4 < 12. We have to determine if x = -3 satisfies both of these inequalities simultaneously.

Step 2: Substituting into the Inequalities (a)

Replace x with -3 in both parts of the inequality 3 < -3x + 4 < 12. This gives us two expressions to evaluate: -1 < -3(-3) + 4 and -3(-3) + 4 < 12.

Step 3: Simplifying the First Inequality (a)

Calculate -3(-3) + 4 which simplifies to 13, resulting in -1 < 13. This inequality is true.

Step 4: Simplifying the Second Inequality (a)

Calculate -3(-3) + 4 which simplifies to 13, resulting in 13 < 12. This inequality is false, so x = -3 is not a solution.

Step 5: Substituting into the Inequalities (b)

Now use the double inequality 3 < -3x + 4 < 14. Replace x with -3, giving: -1 < -3(-3) + 4 and -3(-3) + 4 < 14.

Step 6: Simplifying the First Inequality (b)

For the first part, -3(-3) + 4 simplifies to 13, giving us -1 < 13. This is true.

Step 7: Simplifying the Second Inequality (b)

For the second part, -3(-3) + 4 simplifies to 13, giving us 13 < 14. This is true. Hence, x = -3 is a solution.

Key Concepts

Solution of Double Inequalities

Double inequalities involve two inequalities combined into a single statement. For example, in the inequality \(-1 < -3x + 4 < 12\), you have two parts: \(-1 < -3x + 4\) and \(-3x + 4 < 12\). To find a solution, any potential value for \(x\) must satisfy both conditions simultaneously. If one of the inequalities is false, then the value is not a valid solution.

In the exercise, we're checking if \(x = -3\) is a solution to the inequalities presented. It requires substituting \(-3\) into each part and ensuring that the resulting statements hold true. If both inequalities are true for a given \(x\) value, it is considered a solution.

Substituting Values into Inequalities

To determine if a certain value is a solution for a double inequality, you need to substitute that value into the inequality's expression. Substitution involves replacing the variable with the given number and evaluating the resulting expressions.

For example, in the inequality \(-1 < -3x + 4 < 12\), you replace \(x\) with \(-3\). This substitution transforms the inequality into two separate expressions that need evaluation:

• \(-1 < -3(-3) + 4\)
• \(-3(-3) + 4 < 12\)

Evaluating these will show if \(-3\) satisfies both parts. This process allows us to see whether the inequality holds for that specific value. It's crucial to perform a substitution like this correctly, as errors can lead to incorrect conclusions.

Simplifying Inequalities

Simplification is a key step when dealing with inequalities, helping to clearly see if a value satisfies the inequality. Simplifying involves basic arithmetic operations that reduce expressions to simpler forms.

Consider the expressions formed after substitution:

• \(-1 < -3(-3) + 4\) simplifies to \(-1 < 13\)
• \(-3(-3) + 4 < 12\) simplifies to \(13 < 12\)

By performing calculations like \(-3 \times -3 + 4 = 13\), you can visualize the result and directly see how each simplified inequality stands:

• \(-1 < 13\) is true.
• \(13 < 12\) is false.

In a double inequality, both simplified inequalities must result in true statements for \(x\) to be a solution. If one part is false, then the value does not satisfy the full inequality. Simplifying helps identify true statements quickly and effectively.