Solve the equation and inequality for x.
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
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 to the equation ax - 5 = 18 is x = 23/a.
Part B: Solve the inequality for x showing all steps for credit: ax - 5 > 18
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. However, if a is negative, the direction of the inequality sign must be reversed:
If a > 0:
ax/a > 23/a
x > 23/a
If a < 0:
ax/a < 23/a
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 is either x > 23/a (if a > 0) or x < 23/a (if a < 0).
Part C: The main difference between an equation and an inequality regarding solving steps and solutions is the range of possible solutions.
In an equation, we are looking for a specific value or set of values that satisfy the equation. We can solve it by performing operations on both sides to isolate the variable.
In an inequality, we are looking for a range of values that satisfy the inequality. We can solve it by performing operations on both sides, but we need to consider the direction of the inequality sign and whether the variable may be positive or negative.
Equations have exact solutions, while inequalities have solution sets representing a range or interval of values.
Part A: To solve the equation ax - 5 = 18, we need to isolate the variable x.
1. Start by adding 5 to both sides of the equation.
(ax - 5) + 5 = 18 + 5
ax = 23
2. Next, divide both sides of the equation by a to solve for x.
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 will follow a similar process as in Part A:
1. Start by adding 5 to both sides of the inequality.
(ax - 5) + 5 > 18 + 5
ax > 23
2. Next, divide both sides of the inequality by a. However, if a is negative, we need to flip the inequality sign.
a > 0: ax/a > 23/a
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 the equal sign (=) in an equation and the inequality symbol (<, >, ≤, ≥) in an inequality.
When solving an equation, the goal is to find the value(s) of the variable that make the equation true. To do this, we perform operations on both sides of the equation to isolate the variable.
On the other hand, when solving an inequality, the goal is to find the values of the variable that satisfy the inequality. We perform similar operations on both sides, but we must be mindful of reversing the inequality sign if we multiply or divide by a negative number.
The solutions to an equation are usually expressed as specific values or a set of values, while the solutions to an inequality are usually expressed as a range or interval of values.