Part A. Solve the equation for x showing all steps for credit: ax- 5 = 18
Part B. Solve the inequality for x showing all steps for credit: ax - 5 > 18
Part C: State the difference between an equation and an inequality regarding solving steps and solutions
Part A. Solve the equation for x showing all steps for credit: ax - 5 = 18
To solve this equation for x, we need to isolate the variable x on one side of the equation.
Step 1: Add 5 to both sides of the equation:
(ax - 5) + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a
Therefore, the solution for the equation is x = 23/a.
Part B. Solve the inequality for x showing all steps for credit: ax - 5 > 18
To solve this inequality for x, we need to isolate the variable x on one side of the inequality sign.
Step 1: Add 5 to both sides of the inequality:
(ax - 5) + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by a:
(ax)/a > 23/a
x > 23/a
Therefore, the solution for the inequality is x > 23/a.
Part C: State the difference between an equation and an inequality regarding solving steps and solutions
The main difference between an equation and an inequality is that an equation represents a statement of equality, whereas an inequality represents a statement of inequality.
When solving equations, the goal is to find a specific value that makes the equation true. The solving steps involve performing arithmetic operations to isolate the variable and determine the value that satisfies the equation.
On the other hand, when solving inequalities, the goal is to find a range of values that satisfy the inequality statement. The solving steps involve performing arithmetic operations to isolate the variable and determine the range of values that make the inequality true.
Therefore, the solutions for equations are typically specific values, while the solutions for inequalities are ranges of values.
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x. Here are the steps:
1. Start with the equation: ax - 5 = 18.
2. Add 5 to both sides of the equation to get rid of the constant term on the left side: ax - 5 + 5 = 18 + 5.
Simplifying, we get: ax = 23.
3. To isolate x, divide both sides of the equation by the coefficient of x, which is a: (ax)/a = 23/a.
Simplifying, we get: x = 23/a.
So the solution to the equation is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18, we need to follow similar steps while accounting for the inequality. Here are the steps:
1. Start with the inequality: ax - 5 > 18.
2. Add 5 to both sides of the inequality to get rid of the constant term on the left side: ax - 5 + 5 > 18 + 5.
Simplifying, we get: ax > 23.
3. To isolate x, divide both sides of the inequality by the coefficient of x, which is a: (ax)/a > 23/a.
Remember, when dividing both sides by a negative number, flip the inequality sign. But since a can be positive or negative, we won't assume its sign.
Simplifying, we get: x > 23/a.
So the solution to the inequality is x > 23/a.
Part C:
The main difference between an equation and an inequality is the presence of an equal (=) sign in the equation and a comparison symbol (>, <, ≥, ≤) in the inequality.
When solving an equation, we aim to find the value or values of the variable that make the equation true. The solving steps involve performing operations to isolate the variable on one side of the equation.
In an inequality, we also aim to find the values that satisfy the inequality, but the solution is not limited to a single value. It can be expressed as a range of values or a set of values that satisfy the inequality. Additionally, when dividing or multiplying both sides of an inequality by a negative number, we need to flip the inequality sign.
To solve both equations and inequalities, we follow a series of steps, including adding or subtracting terms, multiplying or dividing both sides, and simplifying to isolate the variable. However, the key difference lies in the nature of the solutions and the way they are expressed.