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, we need to isolate x.
Adding 5 to both sides, we get:
ax - 5 + 5 = 18 + 5
ax = 23
To solve for x, we divide both sides by a:
x = 23/a
Part B:
To solve the inequality ax - 5 > 18, we need to isolate x.
Adding 5 to both sides, we get:
ax - 5 + 5 > 18 + 5
ax > 23
To solve for x, we divide both sides by a:
x > 23/a
Part C:
The difference between an equation and an inequality lies in the range of possible solutions. Equations typically have one unique solution, whereas inequalities can have multiple solutions or a range of values that satisfy the given conditions. When solving an equation, the goal is to find the exact value of the unknown variable. In contrast, when solving an inequality, the goal is to find all values that satisfy the given conditions, often resulting in a range of solutions or a set of values that make the inequality true.
is that suppose to be 23/a or 23 divided by a
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 to isolate the term with x.
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
So, 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 to isolate the term with x.
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by a to solve for x.
ax/a > 23/a
x > 23/a
So, the solution to the inequality ax - 5 > 18 is x > 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions is as follows:
- An equation represents a mathematical equality where two expressions are equal to each other. The goal in solving an equation is to find the value(s) of the variable that satisfy the equation. The solution(s) to an equation is/are typically a specific value or a set of values.
- An inequality represents a mathematical statement showing a relationship between two expressions that are not necessarily equal. In solving an inequality, the goal is to find the values of the variable that make the inequality true. The solution to an inequality is often represented as a range of values or as a set of solutions.
The solving steps for an equation and an inequality are similar, involving operations such as addition, subtraction, multiplication, and division. However, when solving an inequality, it is important to be cautious with the direction of the inequality sign when multiplying or dividing by a negative number, as it may result in flipping the inequality.
Part A: Solve the equation ax - 5 = 18
To solve this equation for x, we need to isolate the variable x on one side of the equation.
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:
(1/a) * ax = (1/a) * 23
x = 23/a
Therefore, the solution for the equation ax - 5 = 18 is x = 23/a.
Part B: Solve the inequality ax - 5 > 18
To solve this inequality for x, we need to isolate the variable x on one side of the inequality.
1. Start by adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
2. If a is positive, we can divide both sides of the inequality by a without changing the direction of the inequality. However, if a is negative, we need to reverse the inequality symbol when dividing.
a > 0:
(1/a) * ax > (1/a) * 23
x > 23/a
a < 0:
(1/a) * ax < (1/a) * 23
x < 23/a
Therefore, depending on the value of a and its relation to 0, the solution for the inequality ax - 5 > 18 can be x > 23/a or x < 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions.
An equation is a statement where two expressions are equal, such as ax - 5 = 18. The goal when solving an equation is to find the values of the variable that make the equation true. The solution to an equation is the value(s) of the variable that satisfy the equation.
On the other hand, an inequality is a statement where two expressions are compared using inequality symbols, such as ax - 5 > 18. The goal when solving an inequality is to find the values of the variable that satisfy the given inequality. The solution to an inequality is a range of values of the variable that make the inequality true.
When solving an equation, the aim is to find a single value (or a set of values) that satisfies the equation. Inequalities, however, have a broader set of solutions since they allow for ranges of values that satisfy the inequality.
The steps for solving equations and inequalities are generally similar, involving operations such as addition, subtraction, multiplication, and division to manipulate the expressions until the variable is isolated on one side. However, when solving inequalities, it's important to consider the direction of the inequality (greater than, less than, greater than or equal to, less than or equal to) and adjust the steps accordingly.