Problem 62 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(9-(x+3) \leq 10\) (a) \(x=-4\) (b) \(x=4\) (c) \(x=0\) (d) \(x=-6\)

Step-by-Step Solution

Answer

The values of x for which the inequality \(9-(x+3) \leq 10\) holds true are x = -4, x = 4, and x = 0. The inequality does not hold true for x = -6.

Step 1: Solution for x = -4

Substitute x = -4 into the inequality: \(9 - ((-4) + 3) \leq 10\), which becomes \(9 - (-1) \leq 10\). Simplify to get \(10 \leq 10\), which is true.

Step 2: Solution for x = 4

Substitute x = 4 into the inequality: \(9 - (4 + 3) \leq 10\), which becomes \(9 - 7 \leq 10\). Simplify to get \(2 \leq 10\), which is true.

Step 3: Solution for x = 0

Substitute x = 0 into the inequality: \(9 - (0 + 3) \leq 10\), which becomes \(9 - 3 \leq 10\). Simplify to get \(6 \leq 10\), which is true.

Step 4: Solution for x = -6

Substitute x = -6 into the inequality: \(9 - ((-6) + 3) \leq 10\), which becomes \(9 - (-3) \leq 10\). Simplify to get \(12 \leq 10\), which is false.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve variables and can be solved much like linear equations, with a few extra rules to consider. These inequalities show the relationship between two expressions and will indicate if one is less than or greater than the other, possibly including equality.

In the given problem, we have the inequality \(9 - (x + 3) \leq 10\). This is an example of a linear inequality that we can solve to find the values of \(x\) that make the inequality true. Unlike linear equations, the solutions to a linear inequality are not just single numbers. Rather, they are ranges or sets of numbers that satisfy the inequality. However, by testing specific values as we see in the exercise, we can determine whether those values belong to the solution set.

Substitution Method

The substitution method is a technique used in algebra to find the solution for variables within equations or inequalities. It involves replacing the variables with their actual values to determine the truth of the given statement.

In our case, we applied the substitution method by replacing the variable \(x\) with the given values -4, 4, 0, and -6, one at a time. Through substitution, you simplify the equation or inequality to find out if the resulting statement is true or false. This method is very efficient for verifying proposed solutions to both equations and inequalities.

Algebraic Solutions

An algebraic solution refers to the process of solving equations or inequalities using algebraic manipulations. These manipulations include operations such as addition, subtraction, multiplication, and division, as well as the application of algebraic properties.

When solving the exercise, we simplified the inequality after substituting the values of \(x\) using arithmetic, demonstrating an algebraic solution. It is important to remember that when you multiply or divide by a negative number while solving an inequality, the inequality sign must be flipped. However, this was not necessary in our example since we only added and subtracted.

Inequality Verification

Inequality verification is the final step of checking whether the values substituted into an inequality hold true or false. After simplifying the inequality with the substituted values, if the statement is logically correct, the value is part of the solution set.

In our exercise, after substituting and simplifying, we verified the inequalities: \(10 \leq 10\), \(2 \leq 10\), and \(6 \leq 10\) are all true, while \(12 \leq 10\) is false. This means that \(x = -4\), \(x = 4\), and \(x = 0\) make the inequality true, but \(x = -6\) does not. Thus, verification is crucial as it confirms the correctness of our solutions.