Solve the Inequality for h 4(3h-7)< =20(h-1)
The given problem is asking to find the values of the variable \( h \) that satisfy the inequality \( 4(3h - 7) \leq 20(h - 1) \). This is a linear inequality which involves simplifying both sides, distributing the numbers to the terms inside the parentheses, and then solving for the unknown variable \( h \) by performing algebraic operations such as addition, subtraction, multiplication, division, and isolating the variable on one side of the inequality. The solution will be a range or set of values that \( h \) can take that makes the inequality true.
$4 \left(\right. 3 h - 7 \left.\right) \leq 20 \left(\right. h - 1 \left.\right)$
Answer
Solution:
Step 1: Expand the left-hand side expression
Step 1.1: Start with the original inequality. Add zero terms if necessary for clarity.
$$0 + 0 + 4(3h - 7) \leq 20(h - 1)$$
Step 1.2: Recognize that adding zero does not change the inequality.
$$4(3h - 7) \leq 20(h - 1)$$
Step 1.3: Use the distributive property to expand the terms.
$$4 \cdot 3h + 4 \cdot (-7) \leq 20(h - 1)$$
Step 1.4: Perform the multiplication.
Step 1.4.1: Multiply 3 by 4.
$$12h + 4 \cdot (-7) \leq 20(h - 1)$$
Step 1.4.2: Multiply 4 by -7.
$$12h - 28 \leq 20(h - 1)$$
Step 2: Expand the right-hand side expression
Step 2.1: Apply the distributive property.
$$12h - 28 \leq 20h + 20 \cdot (-1)$$
Step 2.2: Multiply 20 by -1.
$$12h - 28 \leq 20h - 20$$
Step 3: Isolate terms with h on one side
Step 3.1: Subtract 20h from both sides to move h terms to the left.
$$12h - 28 - 20h \leq -20$$
Step 3.2: Combine like terms.
$$-8h - 28 \leq -20$$
Step 4: Isolate constant terms on the other side
Step 4.1: Add 28 to both sides to move constants to the right.
$$-8h \leq -20 + 28$$
Step 4.2: Combine the constants.
$$-8h \leq 8$$
Step 5: Solve for h
Step 5.1: Divide by -8, remembering to reverse the inequality.
$$\frac{-8h}{-8} \geq \frac{8}{-8}$$
Step 5.2: Simplify both sides.
Step 5.2.1: Cancel out the -8 on the left.
Step 5.2.1.1: Cancel the common factors.
$$\frac{\cancel{-8}h}{\cancel{-8}} \geq \frac{8}{-8}$$
Step 5.2.1.2: Simplify to h.
$$h \geq \frac{8}{-8}$$
Step 5.3: Simplify the right side.
Step 5.3.1: Divide 8 by -8.
$$h \geq -1$$
Step 6: Express the solution in different forms
Inequality Form:
$$h \geq -1$$
Interval Notation:
$$[-1, \infty)$$
Knowledge Notes:
To solve the inequality $4(3h-7) \leq 20(h-1)$, we followed a structured approach:
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions by multiplying each term inside the parentheses by the factor outside.
Combining Like Terms: Terms that contain the same variables to the same power can be combined by adding or subtracting their coefficients.
Inequality Manipulation: When solving inequalities, you can add, subtract, multiply, or divide both sides by the same number. However, when multiplying or dividing by a negative number, the direction of the inequality sign must be reversed.
Interval Notation: This is a way of writing the set of all numbers between two endpoints. The notation $[a, b)$ means the interval includes $a$ and goes up to, but does not include, $b$.
Negative Division: When dividing both sides of an inequality by a negative number, the inequality sign flips direction, which is a critical step in ensuring the correct solution.
By following these steps and applying the relevant mathematical rules, we can solve the inequality and express the solution in various forms.
