Problem 11 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(5 x-12>0\) (a) \(x=3\) (b) \(x=-3\) (c) \(x=\frac{5}{2}\) (d) \(x=\frac{3}{2}\)

Step-by-Step Solution

Answer

x=3 and x=5/2 both make the inequality \(5x-12>0\) true whereas x=-3 and x=3/2 do not.

Step 1: Understand the inequality equation

This step involves understanding the inequality equation given which is \(5x-12>0\). It states that five times a certain number \(x\) minus twelve is greater than zero.

Step 2: Substitution for x=3

Substitute the value of x=3 into the inequality equation, yielding \(5*3-12\). Solve the expression to determine if the result is greater than zero.

Step 3: Substitution for x=-3

Substitute the value of x=-3 into the inequality equation, yielding \(5*-3-12\). Solve the expression to determine if the result is greater than zero.

Step 4: Substitution for x=5/2

Substitute the value of x=5/2 into the inequality equation, yielding \(5*(5/2)-12\). Solve the expression to determine if the result is greater than zero.

Step 5: Substitution for x=3/2

Substitute the value of x=3/2 into the inequality equation, yielding \(5*(3/2)-12\). Solve the expression to determine if the result is greater than zero.

Key Concepts

Substitution Method

The substitution method is a powerful tool in solving mathematical inequalities and equations.By substituting different values for the variable in question, we can check if these values satisfy the given inequality or equation.In the context of this exercise, we are given the inequality \(5x - 12 > 0\).To verify if a particular value of \(x\) is a solution:

• Substitute the value into the inequality.
• Simplify the resulting expression.
• Determine if the expression satisfies the inequality condition.

For example, for \(x = 3\), we substitute \(3\) for \(x\) in the inequality, resulting in \(5 \times 3 - 12 = 15 - 12 = 3\).Since \(3 > 0\), \(x = 3\) is a solution.This step-by-step process allows determining whether a given \(x\) value is within the solution set of the inequality.

Inequality Solutions

Inequality solutions provide the set of all possible values of a variable that satisfy a given inequality.Unlike equations, where solutions are typically exact values, inequalities express a range of values.For the inequality \(5x - 12 > 0\), our task is to find which values satisfy the expression.In practical terms:

• We solve the inequality by isolating \(x\).
• This involves treating it similar to an equation until the inequality sign instead of an equality sign is reached.
• For instance, add \(12\) to both sides: \(5x > 12\).
• And then divide both sides by \(5\): \(x > \frac{12}{5}\) or \(x > 2.4\).

Now, to check if any specific \(x\) such as in our exercise (like \(x = 3\) or \(x = -3\) among others) is a solution, we simply see if it falls within this range.Understanding this approach helps in quickly identifying possible solutions without substituting every value.

Linear Inequalities

Linear inequalities are similar to linear equations but have inequality signs (\(>, <, \geq, \leq)\) instead of the equal sign.They describe a region of the number line or coordinate plane where the inequality holds true.In the given problem \(5x - 12 > 0\), the inequality involves a single variable and is linear because it can be rewritten in the standard linear form without higher powers or products of variables.Consider the properties of linear inequalities:

• They provide a range of solutions rather than specific answers.
• The solution set can often be visualized on a number line as a segment or a ray.
• Operations on linear inequalities follow similar rules to those on equations, ensuring that any operation does not alter the inequality direction unless multiplying or dividing by a negative number.

Understanding linear inequalities and how they differ from linear equations is key.It helps solve the problems by visualizing potential solutions and checking them efficiently.