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Math  /  Algebra

QuestionWRITING Compare solving multi-step inequalities and solving mult-step equation. WRITING Without solving, how can you tell that the inequality 4x+8≤4x−34 x+8 \leq 4 x-3 has no solu

Studdy Solution

STEP 1

1. We need to compare the process of solving multi-step inequalities with solving multi-step equations.
2. We must analyze a given inequality, 4x+8≤4x−34x + 8 \leq 4x - 3, to determine if it has any solutions without solving it.

STEP 2

1. Compare the general process of solving multi-step inequalities and multi-step equations.
2. Examine the given inequality to determine if it has any solutions without solving it.

STEP 3

Identify the steps involved in solving multi-step equations.

STEP 4

Identify the steps involved in solving multi-step inequalities.

STEP 5

Compare the steps of solving multi-step equations with those of solving multi-step inequalities, noting the key differences.

STEP 6

Rewrite the given inequality 4x+8≤4x−34x + 8 \leq 4x - 3 and compare the terms on both sides.

STEP 7

Observe that the terms involving xx are the same on both sides of the inequality, i.e., 4x4x on both sides.

STEP 8

Consider the constant terms on both sides: 88 on the left side and −3-3 on the right side.

STEP 9

Recognize that 88 is not less than or equal to −3-3, implying that no value of xx will satisfy the inequality 4x+8≤4x−34x + 8 \leq 4x - 3.
STEP_1: Solving multi-step equations typically involves the following steps:
1. Simplifying both sides (e.g., combining like terms)
2. Using the properties of equality to isolate the variable (e.g., adding, subtracting, multiplying, or dividing both sides by the same number)
3. Solving for the variable and checking the solution.
STEP_2: Solving multi-step inequalities involves similar steps to solving multi-step equations but with important differences:
1. Simplifying both sides (e.g., combining like terms)
2. Using the properties of inequalities to isolate the variable (e.g., adding, subtracting, multiplying, or dividing both sides by the same number)
3. Solving for the variable and checking the solution
4. Paying special attention to the direction of the inequality when multiplying or dividing by a negative number (the inequality direction reverses).
STEP_3: Key differences between solving multi-step equations and inequalities include:
1. When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality must be reversed.
2. Inequalities often result in a range of solutions, while equations typically yield a single solution or a finite set of solutions.
3. Graphical representation of solutions can differ, with inequalities often being represented as intervals on a number line.
STEP_4: Consider the given inequality: 4x+8≤4x−3 4x + 8 \leq 4x - 3
STEP_5: Observe that the terms involving xx are the same on both sides of the inequality: 4x+8≤4x−3 4x + 8 \leq 4x - 3
Subtracting 4x4x from both sides, we get: 8≤−3 8 \leq -3
STEP_6: Since 88 is not less than or equal to −3-3, the inequality 8≤−38 \leq -3 is false. Therefore, the original inequality has no solutions.
Solution: The inequality 4x+8≤4x−34x + 8 \leq 4x - 3 has no solutions.

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