Solve the equation and inequality for x.
Part A: Solve the equation showing all steps.
ax−5=18
Part B: Solve the inequality showing all steps.
ax−5>18
Answer these Questions:
1. Describe the number line set up that would show all of the solutions for the inequality in Part B.
2. State the difference between solving an equation and an inequality
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x. Let's go through the steps:
1. Add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
2. Divide both sides of the equation by a:
(ax)/a = 23/a
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 isolate the variable x. Let's go through the steps:
1. Add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
2. Divide both sides of the inequality by a. We need to be cautious here since we don't know the value of a. If a is positive, we can divide by a, and if a is negative, we need to flip the inequality sign:
Case 1: a > 0
(ax)/a > 23/a
x > 23/a
Case 2: a < 0
(ax)/a < 23/a (we flip the inequality sign since we divided by a negative number)
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 is either x > 23/a (when a > 0) or x < 23/a (when a < 0).
Answer to Questions:
1. The number line set up that would show all of the solutions for the inequality in Part B would have an open circle at 23/a (when a > 0) and an open circle at 23/a (when a < 0), with an arrow extending to the right (when a > 0) or to the left (when a < 0).
2. The difference between solving an equation and an inequality is that when solving an equation, we are looking for a specific value or values that make the equation true, while when solving an inequality, we are looking for a range of values that satisfy the inequality. In an equation, we often have an equal sign, and in an inequality, we have inequality signs such as >, <, ≥, or ≤.
Part A: Solve the equation showing all steps.
ax - 5 = 18
Step 1: Add 5 to both sides to isolate the variable:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides by a to solve for x:
(ax)/a = 23/a
x = 23/a
Therefore, the solution to the equation ax - 5 = 18 is x = 23/a.
Part B: Solve the inequality showing all steps.
ax - 5 > 18
Step 1: Add 5 to both sides to isolate the variable:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides by a, remembering to consider the case when a is negative (then the inequality sign should be flipped):
For a > 0:
(ax)/a > 23/a
x > 23/a
For a < 0:
(ax)/a < 23/a
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 is:
1. For a > 0, x > 23/a.
2. For a < 0, x < 23/a.
Answer to Question 1:
To show all the solutions for the inequality ax - 5 > 18 on a number line, you would need to create a closed circle at 23/a (for a > 0) or an open circle at 23/a (for a < 0) and draw an arrow to the right (for a > 0) or to the left (for a < 0) from the circle.
Answer to Question 2:
The difference between solving an equation and an inequality is that when solving an equation, we find the specific value(s) of the variable that make the equation true. In contrast, when solving an inequality, we find the range of values for the variable that satisfy the inequality, which may involve multiple solutions.