⚖️ What's the main difference between solving equations and solving inequalities?

<p class="font-claude-response-body break-words whitespace-normal" dir="ltr">When solving equations, you use the same operations on both sides to find one specific value. With inequalities, you do the same—add, subtract, multiply, or divide both sides the same way. The key difference: when you <strong>multiply or divide by a negative number, you must flip the inequality sign</strong>. So an equation gives you one answer, but an inequality gives you a range of values.</p>

📍 How do I know when to use an open circle versus a closed circle on a number line?

<div role="feed" aria-label="Chat messages" aria-describedby="_r_1k0_" aria-busy="false" data-find-provider-scope=""> <div data-sizer-excess="0" data-rocksteady-sizer=""> <div data-rs-index="601" data-index="601" data-last-message="true"> <div tabindex="0" role="article" aria-setsize="602" aria-posinset="602" aria-label="Message 602 of 602"> <div data-test-render-count="1"> <div class="group group/message-row"> <div class="contents"> <div class="group relative relative pb-[var(--msg-assistant-pb,0.75rem)]" data-is-streaming="false"> <div class="font-claude-response relative leading-[1.65rem] [&_pre>div]:bg-bg-000/50 [&_pre>div]:border-0.5 [&_pre>div]:border-border-400 [&_.ignore-pre-bg>div]:bg-transparent [&_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8"> <div> <div class="grid grid-rows-[auto_auto] min-w-0"> <div class="row-start-2 col-start-1 relative grid grid-rows-[auto_auto] isolate min-w-0"> <div class="row-start-1 col-start-1 relative z-[2] min-w-0"> <div> <div> <div class="standard-markdown grid-cols-1 grid [&_>_*]:min-w-0 gap-3 [&_>_*:last-child]:mb-0 print:block print:[&_>_*_+_*]:mt-3 standard-markdown"> <p class="font-claude-response-body break-words whitespace-normal" dir="ltr">Use a <strong>closed circle when the number is included in the solution</strong>. This happens with "less than or equal to" and "greater than or equal to" signs. Use an <strong>open circle when the number is not included</strong>, which occurs with "less than" and "greater than" only.</p> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div>

Understanding and Solving Inequalities

An inequality is an algebraic comparison in which its two members are linked by one of these signs:

$\text{[math]}$

The solution of an inequality is the set of values of the variable that satisfy the inequality.

We can express the solution of the inequality through a graphical representation or an interval.

Examples

1. Solve the inequality 2x - 1 <7

2x-1<7

2x<8

x<4

Graphical representation: Representación gráfica del intervalo abierto de menos infinito a cuatro

Interval: $\text{[math]}$

2. Solve the inequality $\text{[math]}$

$\text{[math]}$

Graphical representation: Representación gráfica del intervalo cerrado de menos infinito a cuatro

Interval: $\text{[math]}$

3. Solve the inequality 2x - 1 > 7

2x-1>7

2x>8

x>4

Graphical representation: Representación gráfica del intervalo abierto de cuatro a infinito

Interval: $\text{[math]}$

4. Solve the inequality $\text{[math]}$

$\text{[math]}$

Graphical representation: Representación gráfica del intervalo cerrado de cuatro a infinito

Interval: $\text{[math]}$

Equivalence Criteria for Inequalities

Property 1: If the same number is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original.

3x+4<5

3x+4-4<5-4

3x<1

Property 2: If both sides of an inequality are multiplied or divided by the same positive number, the resulting inequality is equivalent to the original.

2x<6

2x\div 2<6\div 2

x<3

Property 3: If both sides of an inequality are multiplied or divided by the same negative number, the resulting inequality reverses direction and is equivalent to the original.

-x<5

-x\cdot (-1)>5\cdot (-1)

x>-5

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