Problem 60 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(3(x+5)-4>2\) (a) \(x=3\) (b) \(x=0\) (c) \(x=-4\) (d) \(x=-10\)

Step-by-Step Solution

Answer

Among the given values, \(x=3\) and \(x=0\) are solutions to the inequality, while \(x=-4\) and \(x=-10\) are not.

Step 1: Simplify the inequality

To simplify the inequality, distribute the 3 in \(3(x+5)-4>2\), and then isolate \(x\). That gives: \(3x + 15 - 4 > 2\) which simplifies to: \(3x + 11 > 2\)Therefore, the isolated inequality is: \(3x > -9\) or \(x > -3\).

Step 2: Test \(x=3\)

Substitute \(x=3\) into the inequality \(x > -3\). This gives: \(3 > -3\), which is true. So, \(x=3\) is a solution of the inequality.

Step 3: Test \(x=0\)

Substitute \(x=0\) into the inequality \(x > -3\). This gives: \(0 > -3\), which is true. So, \(x=0\) is a solution of the inequality.

Step 4: Test \(x=-4\)

Substitute \(x=-4\) into the inequality \(x > -3\). This gives: \(-4 > -3\), which is not true. So, \(x=-4\) is not a solution of the inequality.

Step 5: Test \(x=-10\)

Substitute \(x=-10\) into the inequality \(x > -3\). This gives: \(-10 > -3\), which is not true. So, \(x=-10\) is not a solution of the inequality.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations, like addition, subtraction, multiplication, and division. In this problem, you see an expression like \(3(x+5)-4\), which is a typical algebraic expression. Here, \(x\) is the variable and can take on different values. When you are faced with simplifying these kinds of expressions, one approach is to distribute any multiplication over addition or subtraction first.

For example, in the expression \(3(x + 5)\), you distribute the 3 into both \(x\) and 5, making the expression \(3x + 15\). This is known as using the distributive property. After simplifying, you often end up with something that is much easier to work with or solve. Knowing how to manipulate these expressions is crucial in algebra.

Solving Inequalities

Solving inequalities involves finding all possible values for a variable that make the inequality true. Inequalities are similar to equations, but instead of an equal sign, they use signs like \(>\), \(<\), \(\geq\), or \(\leq\).

For the inequality \(3x + 11 > 2\), you solve it much like you would for an equation. First, you isolate the variable by performing operations on both sides of the inequality:

• Subtract 11 from both sides: \(3x + 11 - 11 > 2 - 11\) gives \(3x > -9\).
• Then, divide by 3 to isolate \(x\): \(x > -3\).

Remember that if you multiply or divide by a negative number, you must flip the inequality sign. Understanding these rules ensures you solve inequalities correctly, finding the range of values that satisfy the condition.

Mathematical Reasoning

Mathematical reasoning is the process of drawing logical conclusions based on given facts or premises. In solving inequalities, mathematical reasoning is used when deciding if a certain value of \(x\) makes the inequality true or false.

After isolating \(x\) in the inequality to find \(x > -3\), you need to determine if each given \(x\) is a solution. Substituting each value of \(x\) into the inequality is a good way to test this:

• For \(x = 3\), substituting gives \(3 > -3\), which is true.
• For \(x = 0\), substituting gives \(0 > -3\), also true.
• For \(x = -4\) or \(x = -10\), these do not hold as \(-4 > -3\) and \(-10 > -3\) are false.

This logical reasoning step ensures you're accurately interpreting and solving the inequality by checking each hypothesis (value of \(x\)) against the previously established inequality condition. Reasoning helps confirm if your solution process and answer are correct and aligns with the problem given.