16. 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
To solve the equation, we need to isolate x.
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 to solve for x:
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
To solve the inequality, we follow similar steps as in Part A, but with one additional step.
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 to solve for x. However, if a is negative, we need to flip the inequality symbol as well:
x > 23/a (if a > 0)
x < 23/a (if a < 0)
So the solution to the inequality ax - 5 > 18 is x > 23/a or x < 23/a, depending on the value of a.
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.
An equation is a statement that two expressions are equal, while an inequality is a statement that two expressions are not equal and one is greater or less than the other.
When solving an equation, the goal is to find the value(s) of the variable that make the equation true. The steps involved usually include isolating the variable on one side of the equation.
When solving an inequality, the goal is to find the value(s) of the variable that make the inequality true. The steps involved are similar to solving an equation but with an additional consideration for the direction of the inequality symbol. Inequalities can have multiple solutions and can often be represented by a range of values.
In summary, the main difference is that equations focus on finding equality, while inequalities focus on finding different relationships, such as greater than or less than.
Part A: To solve the equation ax - 5 = 18, you need to isolate x. Follow these steps:
1. Begin by adding 5 to both sides of the equation: ax - 5 + 5 = 18 + 5
This simplifies to: ax = 23
2. Next, divide both sides of the equation by a to solve for x: (ax) / a = 23 / a
This simplifies to: x = 23/a
So the solution to the equation ax - 5 = 18 is x = 23/a.
Part B: To solve the inequality ax - 5 > 18, you also need to isolate x. Follow these steps:
1. Start by adding 5 to both sides of the inequality: ax - 5 + 5 > 18 + 5
This simplifies to: ax > 23
2. Next, divide both sides of the inequality by a, remembering to change the direction of the inequality if a is negative: (ax) / a > 23 / a
If a is positive, the inequality remains the same: x > 23/a
If a is negative, the inequality is reversed: x < 23/a
So the solution to the inequality ax - 5 > 18 is x > 23/a if a is positive, or x < 23/a if a is negative.
Part C: The difference between an equation and an inequality lies in the nature of the solution set and the solving steps required.
For an equation, the objective is to find the value(s) of the variable (in this case, x) that make the equation true. Equations typically have one or more solutions that satisfy the equation. Generally, you use algebraic techniques such as combining like terms, isolating the variable, and performing arithmetic operations to solve an equation.
On the other hand, an inequality aims to find the range of values for the variable (x) that make the inequality true. Inequalities often have an infinite number of solutions, represented by intervals on a number line or in interval notation. To solve an inequality, you use similar techniques of isolating the variable but need to be mindful of reversing the inequality's direction when multiplying or dividing by a negative number.
In summary, equations have specific solutions, whereas inequalities represent a range of possible solutions. The steps to solve them are similar, but solving an inequality also involves considering the direction of the inequality when performing certain operations.