Problem 11 - Elementary and Intermediate Algebra

Question

Use a check to determine whether \(-5\) is a solution of the following equation and inequality. a. \(5(2 x+7)=2 x-4\) b. \(3 x+6 \leq-9\)

Step-by-Step Solution

Answer

-5 is not a solution for the equation (a) but is a solution for the inequality (b).

Step 1: Substitute -5 into the equation

For equation a) \(5(2x+7) = 2x-4\), substitute \(x = -5\) into both sides of the equation. Compute each side separately. Left side: \(5(2(-5) + 7) = 5(-10 + 7) = 5(-3) = -15\). Right side: \(2(-5) - 4 = -10 - 4 = -14\).

Step 2: Compare the two sides of the equation

To determine if \(-5\) is a solution, the left side of the equation must equal the right side. We computed \(-15\) on the left and \(-14\) on the right. Since \(-15 eq -14\), \(-5\) is not a solution.

Step 3: Substitute -5 into the inequality

For inequality b) \(3x + 6 \leq -9\), substitute \(x = -5\) and calculate the left side.Left side: \(3(-5) + 6 = -15 + 6 = -9\).

Step 4: Compare with the inequality bound

The inequality \(3x + 6 \leq -9\) simplifies to checking if \(-9 \leq -9\). Since \(-9\) is equal to \(-9\), the statement is true, and \(-5\) is a solution of the inequality.

Key Concepts

Solving Equations

Solving equations is an essential part of algebra, which involves finding the value or values of variables that make an equation true. An equation is a mathematical sentence that shows that two expressions are equal, such as \(5(2x+7) = 2x-4\). To solve an equation, you need to find the value of the variable that makes both sides of the equation equal.

There are many methods to solve equations, such as:

• Isolating the variable: Rearrange the equation to have the variable on one side and the constants on the other side.
• Balancing both sides: Whatever operation you apply to one side of the equation, you must apply to the other side to maintain equality.
• Using substitution: This can be useful, as it allows you to verify potential solutions by inserting them back into the equation, much like testing the solution \(x = -5\) in our example.

In practice, solving equations often requires a combination of these techniques to simplify each step and solve for the variable efficiently.

Inequalities

Inequalities are similar to equations, but instead of showing equality, they show that one expression is greater or less than another. In the inequality \(3x + 6 \leq -9\), the symbol "\(\leq\)" indicates that the left side is less than or equal to the right side.

When solving inequalities:

• Treat them similarly to equations, with a focus on maintaining the inequality's direction.
• Use the same operations on both sides to isolate the variable, but be careful with multiplication or division by negative numbers, as this will flip the inequality symbol.

For example, when substituting \(x = -5\) into the inequality \(3(-5) + 6 \leq -9\), you compute the left side as \(-9\), which matches the right side of \(-9\). Because the statement \(-9 \leq -9\) is true, \(-5\) satisfies the inequality.

Substitution Method

The substitution method is a powerful tool in algebra for determining whether a potential value is a solution for an equation or inequality. It involves replacing a variable with a specific number and checking whether the equation or inequality holds true.

Steps for using the substitution method effectively:

• Identify the equation or inequality and the potential solution.
• Substitute the given value into the equation or inequality wherever the variable appears.
• Simplify both sides of the equation or inequality to see if they hold true.

In our example, substituting \(-5\) into both the equation \(5(2x + 7) = 2x - 4\) and the inequality \(3x + 6 \leq -9\) involved calculating the left and right sides separately. Comparing the results indicated that \(-5\) was not a solution for the equation but was a valid solution for the inequality. This method not only helps in direct checking but also builds a deeper understanding of how expressions behave with different values.