Solve each equation or inequality. | 10 - 4x | + 1 ≥ 5

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Start by isolating the absolute value expression. Subtract 1 from both sides of the inequality to get: \(| 10 - 4x | \geq 4\).

Recall that for an absolute value inequality \(|A| \geq B\) (where \(B > 0\)), the solution splits into two cases: \(A \geq B\) or \(A \leq -B\).

Apply this to the inequality: \(10 - 4x \geq 4\) or \(10 - 4x \leq -4\).

Solve each inequality separately. For \(10 - 4x \geq 4\), subtract 10 from both sides and then divide by -4, remembering to reverse the inequality sign when dividing by a negative number. For \(10 - 4x \leq -4\), do the same.

Write the solution as the union of the two solution sets found from the inequalities in step 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, you consider two cases: one where the expression inside the absolute value is positive or zero, and one where it is negative, leading to two separate inequalities.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This simplification allows you to apply the definition of absolute value and split the inequality into two cases more easily.

Solving Linear Inequalities

After splitting the absolute value inequality into two linear inequalities, solve each inequality separately by isolating the variable. The solution set is the union of the solutions from both inequalities, representing all values that satisfy the original inequality.