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:
To solve the equation ax - 5 = 18 for x, we need to isolate x on one side of the equation. Here are 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
Therefore, the solution for x is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18 for x, we need to isolate x on one side of the inequality. Here are 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. However, if a is negative, we must flip the inequality sign:
ax/a > 23/a
If a > 0,
x > 23/a
If a < 0,
x < 23/a
Therefore, the solution for x depends on the value of a.
Part C:
An equation is a statement that two expressions are equal, while an inequality is a statement that two expressions are not equal or that one expression is greater/less than another.
When solving equations, our goal is to find the value(s) of the variable that make the equation true. We perform the same operation on both sides of the equation to isolate the variable.
When solving inequalities, our goal is to find the range of values of the variable that make the inequality true. The solving steps depend on the type of inequality (greater than, less than, greater than or equal to, less than or equal to). We must also consider if the variable is being multiplied or divided by a negative number, as it may change the direction of the inequality sign.
In terms of solutions, an equation can have one solution, no solutions, or infinitely many solutions, while an inequality can have a range of possible solutions, which can be written in interval notation or using inequality symbols.
Part A: Solve the equation ax-5=18
Step 1: Add 5 to both sides of the equation to isolate the variable:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by 'a' to solve for x:
(ax)/a = 23/a
x = 23/a
The solution for the equation ax-5=18 is x = 23/a.
Part B: Solve the inequality ax-5 > 18
Step 1: Add 5 to both sides of the inequality to isolate the variable:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by 'a' to solve for x. Note that if 'a' is negative, the sign of the inequality will flip:
(ax)/a > 23/a
If 'a' is positive:
x > 23/a
If 'a' is negative:
x < 23/a
So, the solution for the inequality ax-5 > 18 is x > 23/a or x < 23/a, depending on the value of 'a'.
Part C: The main difference between an equation and an inequality is in the nature of the solutions.
- Equations: The goal is to find a specific value or values that satisfy the equation. The solutions are often expressed as a specific value or a set of values.
- Inequalities: The goal is to find a range of values that satisfy the inequality. The solutions are expressed as a range or inequality statement, such as x > 5 or x < 10. Inequalities also have the possibility of having an infinite number of solutions, depending on the given conditions.